The Canny–Emiris Conjecture for the Sparse Resultant

Let \(C\in S(\rho )\). Then,

Let \(v \in N_{{\mathbb {R}}}\) and set for short \(\kappa =h_{{\text {epi}}(\rho )}(v,1)\) and \(\kappa _{i}=h_{{\text {epi}}(\rho _{i})}(v,1)\) for each i. We have that \( {\text {epi}}(\rho )= \sum _{i=0}^{s}{\text {epi}}(\rho _{i}) \) and so, by the additivity of the support function,Let \(i\in \{0,\dots , s\}\) and \(x_{i}\in C_{i}\). Choose \(x_{j}\in C_{j}\), \(j\ne i\), such that \((x_{0},\dots , x_{s})\in \Pi _{C}\) and set \(x=\sum _{j=0}^{s}x_{j}\), so that \(x\in C\) and \(\rho (x)=\sum _{j=0}^{s}\rho _{j}(x_{j})\). Hence, \((x,\rho (x))\in {\text {epi}}(\rho )^{(v,1)}\) and \((x_{j},\rho _{j}(x_{j}))\in {\text {epi}}(\rho _{j})\) for all j and soThus, \(\langle (v,1),(x_{i},\rho _{i}(x_{i}))\rangle = \kappa _{i}\) or equivalently \((x_{i},\rho _{i}(x_{i}))\in {\text {epi}}(\rho _{i})^{(v,1)}\), which implies that \(x_{i}\in \Gamma (\rho _{i},v)\).

Conversely, let \(x_{i}\in \Gamma (\rho _{i},v)\). Choose \(x_{j}\in \Gamma (\rho _{j},v)\), \(j\ne i\), and set \(x=\sum _{j=0}^{s}x_{j}\) and \(t=\sum _{j=0}^{s}\rho _{j}(x_{j})\). We have that \(t\ge \rho (x)\) and so \((x,t)\in {\text {epi}}(\rho )\). MoreoverHence, \((x,t)\in {\text {epi}}(\rho )^{(v,1)}\) and so \(x\in C\) and \(t=\rho (x)\). In particular, \(x_{i} \in C_{i}\) and we conclude that \(C_{i}=\Gamma (\rho _{i},v)\), proving (1).

Now let \(x_{i}\in C_{i}\), \(i=0,\dots , s\), and set \(x=\sum _{i=0}^{s}x_{i}\). By (1), for any \(v\in N_{{\mathbb {R}}}\) such that \(C=\Gamma (\rho ,v)\) we have that \(C_{i}=\Gamma (\rho _{i},v)\), and the last part of the proof of this statement shows that \(x\in C\) and \(\rho (x)=\sum _{i=0}^{s}\rho _{i}(x_{i})\). This proves both that \( \sum _{i=0}^{s}C_{i}\subset C\) and the “if” part in (3).

Conversely, for each \(x\in C\) the infimum in (2.5) is attained, and so there are \(x_{i}\in C_{i}\), \(i=0,\dots , s\), with \(x=\sum _{i=0}^{s}x_{i}\). Hence, \(C=\sum _{i=0}^{s}C_{i}\) as stated in (2), whereas the “only if” part in (3) is immediate from the definition of the components in (2.7). \(\square \)

Let \(C, C'\in S(\rho )\) and \(i\in \{0,\dots , s\}\) such that their respective i-th components have both dimension n and coincide. Then, \(C=C'\).

Since both \(C_{i}\) and \(C_{i}'\) have dimension n and coincide, there is a unique \(v\in N_{{\mathbb {R}}}\) with \( C_{i}=C_{i}'=\Gamma (\rho _{i},v)\). Proposition 2.1(1) then implies that \(C=\Gamma (\rho ,v)=C'\). \(\square \)

A mixed subdivision \(S(\rho )\) on \(\Delta \) is tight if for every n-cell C of \( S(\rho )\),If this condition holds, when \(s=n-1\) an n-cell of \(S(\rho )\) is mixed if all its components are segments and when \(s=n\), for \(k=0,\dots , n\) an n-cell of \(S(\rho )\) is k-mixed if its i-th component is a segment for all \(i\ne k\) (and so its k-th component is a point).

An incremental chain of mixed subdivisions of \(\Delta \) is a chain \(S(\theta _{0})\preceq \dots \preceq S(\theta _{s})\) where, for \(k=0,\dots , s \), the mixed subdivision \(S(\theta _{k})\) is induced by the inf-convolution \( \theta _{k}:\Delta \rightarrow {\mathbb {R}}\) of a family of convex piecewise affine functions \(\theta _{k,i} :\Delta _{i}\rightarrow {\mathbb {R}}\), \(i=0,\dots , s\), such that \(\theta _{k,i}= 0|_{\Delta _{i}}\) for \(i\ge k\).

This incremental chain is tight if, for each k, the mixed subdivision of \(\Delta \) (considered as the sum of the \(k+1\) polytopes \(\Delta _0, \dots , \Delta _{k-1}, \sum _{{i=k}}^{s} \Delta _{i}\)) induced by the convex piecewise affine functionsis tight in the sense of Definition 2.3.

The notion of incremental chain of mixed subdivisions of \(\Delta \) might be easily extended to chains of arbitrary length. We have chosen to restrict it to chains of length equal to the number of polyhedra \(\Delta _{i}\) because it is the only case of interest for the proof of Theorem 4.27.

If \( S(\theta _{0})\preceq \dots \preceq S(\theta _{s})\) is a tight incremental chain, then \(S(\theta _{s})\) is a tight mixed subdivision.

Let \(n=2\) and \(M={\mathbb {Z}}^{2}\). Set \(d_{0}=1\), \(d_{1}=3\) and \(d_{2}=2\) and for \(i=0,1,2\) consider the triangle \(\Delta _{i}=\{(x_{1},x_{2})\in ({\mathbb {R}}_{\ge 0})^{2}\mid x_{1}+x_{2}\le d_{i}\}\). Consider also the affine functions \(\rho _{i}:\Delta _{i}\rightarrow {\mathbb {R}}\), \(i=0,1\), defined byFor \(k, i=0,1,2\) set \(\theta _{k,i}=\rho _{i}\) if \(i<k\) and \(\theta _{k,i}=0|_{\Delta _{i}}\) if \(i\ge k\), and then for \(k=0,1,2\) set \(\theta _{k}=\theta _{k,0}\boxplus \theta _{k,1}\boxplus \theta _{k,2}\). Hence, \(S(\theta _{0}) \preceq S(\theta _{1}) \preceq S(\theta _{2}) \) is a tight incremental chain of mixed subdivisions of the triangle \(\Delta =\{(x_{1},x_{2})\in ({\mathbb {R}}_{\ge 0})^{2}\mid x_{1}+x_{2}\le 6\}\) (Fig. 1).

There is a finite union of hyperplanes \(W\subset {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) not containing \(T_{{\varvec{{{\mathcal {A}}}}}}(N^{s+1})\) such that for all \({\varvec{\nu }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}{\setminus } W\) the mixed subdivision \(S(\Theta _{{\varvec{\nu }}})\) of \(\Delta \) is tight.

For \({\varvec{\nu }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) let \({\widehat{L}}\subset M_{{\mathbb {R}}}\times {\mathbb {R}}^{s+1} \times {\mathbb {R}}\) denote the linear span of the vectors \( \{(a, e_{i},\nu _{i,a})\}_{i\in \{0,\dots , s\}, a\in {{\mathcal {A}}}_{i}}\). Set also \({\widehat{{{\mathcal {A}}}}}_{i}= \{(a,\nu _{i,a})\}_{a\in {{\mathcal {A}}}_{i}} \subset M_{{\mathbb {R}}}\times {\mathbb {R}}\), \(i=0,\dots , s\). Then,

For each i let \({\widehat{L}}_{i}\subset M_{{\mathbb {R}}}\times {\mathbb {R}}\) denote the affine span of the nonempty finite subset \({\widehat{{{\mathcal {A}}}}}_{i}\). Making linear combinations between the generators of the linear subspace \({\widehat{L}}\), we easily deduce that \( \dim ({\widehat{L}})= \dim (\sum _{i=0}^{s}{\widehat{L}}_{i}) +s+1\). The statement then follows from the fact that \( \dim (\sum _{i=0}^{s}{\widehat{L}}_{i}) = \dim (\sum _{i=0}^{s}{\text {conv}}({\widehat{{{\mathcal {A}}}}}_{i}))\). \(\square \)

For \(i=0,\dots , s\) let \({\varvec{u}}_{i}\) be a set of \(\# {{\mathcal {A}}}_{i}\) variables and let \({\varvec{u}}=({\varvec{u}}_{0}, \dots , {\varvec{u}}_{s})\). Fix an isomorphism \(M_{{\mathbb {R}}}\simeq {\mathbb {R}}^{n}\). Then, for each family \({\varvec{{{\mathcal {D}}}}}=({{\mathcal {D}}}_{0},\dots , {{\mathcal {D}}}_{s})\) of nonempty subsets \({{\mathcal {D}}}_{i}\subset {{\mathcal {A}}}_{i}\) satisfying the conditions set \( {{\mathcal {G}}}_{{\varvec{{{\mathcal {D}}}}}}\in {\mathbb {R}}[{\varvec{u}}]^{ {\varvec{{{\mathcal {D}}}}}\times (n+s+2)} \) for the square matrix made of the row vectorsSet also \( G_{{\varvec{{{\mathcal {D}}}}}}=\det ({{\mathcal {G}}}_{{\varvec{{{\mathcal {D}}}}}})\in {\mathbb {R}}[{\varvec{u}}]\), which is a linear form.

The conditions on \({\varvec{{{\mathcal {D}}}}}\) imply that for each i there is \({\varvec{\nu }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) such that, settingwe have that \( \dim ( \sum _{i=0}^{s}{\text {conv}}({\widehat{{{\mathcal {D}}}}}_{i})) = n+1 \). Moreover, this condition can be fulfilled with a vector \({\varvec{\nu }}\in T_{{\varvec{{{\mathcal {A}}}}}}(N^{s+1})\). Indeed, the condition (2) implies that the \({\mathcal {D}}_{i}\)’s are simplexes which by the condition (1), have dimensions that sum up \(n+1\). Assuming without loss of generality that each \({\mathcal {D}}_{i}\) contains 0 as one its points, the remaining points in the union of these sets determine \(n+1\) directions which by condition (3) span \(M_{{\mathbb {R}}}\). Then, a possible choice for \({\varvec{\nu }}\) satisfying (2.11) consists in setting \(\nu _{i,a}=0\) for each i and all \(a\in {\mathcal {D}}_{i}\) except one of them, for which this coordinate is set to 1.

Lemma 2.9 then implies \( G_{{\varvec{{{\mathcal {D}}}}}}({\varvec{\nu }})\ne 0\) and in particular, the zero set of \(G_{{\varvec{{{\mathcal {D}}}}}}\) is a hyperplane not containing the linear subspace \(T_{{\varvec{{{\mathcal {A}}}}}}(N^{s+1})\). The set W is then defined as the union of all these hyperplanes.

Now let \({\varvec{\nu }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}} \) and suppose that \(S(\Theta _{{\varvec{\nu }}})\) is not tight. Let C be an n-cell of this mixed subdivision such that \( \sum _{i=0}^{s}\dim (C_{i})>n\). Then, we can choose nonempty finite subsets \({{\mathcal {D}}}_{i} \subset {{\mathcal {A}}}_{i}\cap C_{i}\), \(i=0,\dots , s\), satisfying the conditions (1), (2) and (3). Such a choice may be accomplished by picking simplexes defined by points in the finite subsets \( {{\mathcal {A}}}_{i}\cap C_{i}\), \(i=0,\dots , s\), whose dimensions sum up \(n+1\).

Let P be the face of the graph of \(\Theta _{{\varvec{\nu }}}\) corresponding to C, and for each i let \(P_{i}\) be the face of the graph of \(\vartheta _{{\varvec{\nu }}_{i}}\) corresponding to the component \(C_{i}\). For each i the lifted set \({\widehat{{{\mathcal {D}}}}}_{i}\) as in (2.11) is contained in \( P_{i}\) and soHence, \( \dim ( \sum _{i=0}^{s} {\text {conv}}({\widehat{{{\mathcal {D}}}}}_{i})) \le \dim (P)=\dim (C) = n\). Lemma 2.9 then implies that \(G_{{\varvec{{{\mathcal {D}}}}}}({\varvec{\nu }})=0\) and so \({\varvec{\nu }}\in W\), concluding the proof. \(\square \)

There is a finite union of hyperplanes \(W'\subset {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}'}\) not containing \(T_{{\varvec{{{\mathcal {A}}}}}'}(N^{s})\) such that for all \({\varvec{\nu }}' \in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}'}{\setminus } W'\) the mixed subdivision \(S(\Theta _{{\varvec{\nu }}})\) of \(\Delta \) associated with the vector \({\varvec{\nu }}=({\varvec{\nu }}', {\varvec{0}})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) is tight.

With notation as in Proposition 2.8, choose a vector \((w_{0},\dots , w_{s})\in N^{s+1}\) whose image with respect to the linear map \(T_{{\varvec{{{\mathcal {A}}}}}}\) does not lie in W. Let \({\varvec{\zeta }}_{i} =(\langle w_{s},a\rangle )_{a\in {{\mathcal {A}}}_{i}}\in {\mathbb {R}}^{{{\mathcal {A}}}_{i}}\), \(i=0,\dots , s\), andwhich is a finite union of hyperplanes of \({\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}'}\). We have thatHence, \(T_{{\varvec{{{\mathcal {A}}}}}'}(w_{0}-w_{s}, \dots , w_{s-1}-w_{s}) \notin W'\), and so \(W' \not \supset T_{{\varvec{{{\mathcal {A}}}}}'}(N^{s-1})\).

By Proposition 2.8, for \({\varvec{\nu }}' =({\varvec{\nu }}' _{0}, \dots , {\varvec{\nu }}' _{s-1})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}'}{\setminus } W'\) the mixed subdivision associated with \( ({\varvec{\nu }}'_{0}+ {\varvec{\zeta }}_{0},\dots , {\varvec{\nu }}'_{s-1}+ {\varvec{\zeta }}_{s-1}, {\varvec{\zeta }}_{s})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) is tight. Hence, this is also the case for the mixed subdivision associated with the vector \(({\varvec{\nu }}',{\varvec{0}}) \in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\), since the corresponding functions differ by a globally defined linear one. \(\square \)

Let \({\varvec{\nu }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\). There is a neighborhood U of \({\varvec{\nu }}\) such that for all \({\widetilde{{\varvec{\nu }}}}\in U \) we have that \(S(\Theta _{{\widetilde{{\varvec{\nu }}}}})\succeq S(\Theta _{{\varvec{\nu }}})\).

For each n-cell C of \(S(\Theta _{{\varvec{\nu }}})\) denote by \(v_{C}\) the unique vector in \(N_{{\mathbb {R}}}\) such that \(C=\Gamma (\Theta _{{\varvec{\nu }}}, v_{C})\). By Proposition 2.1(1) and the inequality in (2.4), for each i there is \(\kappa _{C,i}\in {\mathbb {R}}\) such that, for \(x\in \Delta _{i}\),with equality if and only if \(x \in C_{i}\). Hence, there is \(c>0\) such that for all \(a\in {{\mathcal {A}}}_{i}{\setminus } C_{i}\),Let \(\varepsilon >0\) and \({\widetilde{{\varvec{\nu }}}}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}} \) with \(\Vert {\widetilde{{\varvec{\nu }}}}-{\varvec{\nu }}\Vert _{\infty }<\varepsilon \), where \(\Vert \cdot \Vert _{\infty }\) denotes the \(\ell ^{\infty }\)-norm of \({\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\). Then, for all i and \(x\in \Delta _{i}\) we have thatFix a norm \(\Vert \cdot \Vert \) on \(M_{{\mathbb {R}}}\) and let \(\Vert \cdot \Vert ^{\vee }\) be the corresponding operator norm on \(N_{{\mathbb {R}}}\), so that for \(v\in N_{{\mathbb {R}}}\) and \(x\in M_{{\mathbb {R}}}\) we have thatLet \({\widetilde{C}}\) be an n-cell of \(S(\Theta _{{\widetilde{{\varvec{\nu }}}}})\) and, similarly as before, denote by \(v_{{\widetilde{C}}} \in N_{{\mathbb {R}}}\) and \(\kappa _{{\widetilde{C}}, i} \in {\mathbb {R}}\), \(i=0,\dots , s\), the corresponding vector and constants. Then, for each n-cell C of \(S(\Theta _{{\varvec{\nu }}})\) with \(\dim ({\widetilde{C}} \cap C)=n\) there is \(K>0\) such thatSince the number of possible pairs \(({\widetilde{C}}, C)\) for varying \({\widetilde{{\varvec{\nu }}}}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) is finite, the constant \(K>0\) can be taken independently of the choice of these n-cells.

From the inequalities in (2.12) and (2.13), we deduce that for all \(a\in {{\mathcal {A}}}_{i}{\setminus } C_{i}\),and from the inequalities in (2.14) and (2.15),Then, for \(\varepsilon >0\) sufficiently small we have that \( \vartheta _{{\widetilde{{\varvec{\nu }}}}_{i}}(a) >\langle -v_{{\widetilde{C}}},a\rangle +\kappa _{{\widetilde{C}},i} \) for all \(a\in {{\mathcal {A}}}_{i}{\setminus } C_{i}\), which implies that \({\widetilde{C}}_{i}\subset C_{i}\) for all i. In turn, by Proposition 2.1(2) this implies that \({\widetilde{C}}\subset C\).

Since this holds for every n-cell of \(S(\Theta _{{\widetilde{{\varvec{\nu }}}}})\), we deduce that this mixed subdivision refines \(S(\Theta _{{\varvec{\nu }}})\). The statement follows by taking U as the ball of \({\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) centered at \({\varvec{\nu }}\) of radius \(\varepsilon \) with respect to the \(\ell ^{\infty }\)-norm. \(\square \)

Set \({{\mathcal {A}}}_{> i }=\sum _{j> i} {{\mathcal {A}}}_{j} \subset M\), \(i=0,\dots , s-1\), and for each i denote by \( W_{i}'\) the finite union of hyperplanes given by Corollary 2.10 applied to the family \(({{\mathcal {A}}}_{0}, \dots , {{\mathcal {A}}}_{i}, {{\mathcal {A}}}_{> i })\) of \(i+{2}\) nonempty subsets of M.

For \(i=0,\dots , s-1\) choose iteratively \({\varvec{\nu }}_{i} \!\in \! {\mathbb {R}}^{{{\mathcal {A}}}_{i}}\) such that \(({\varvec{\nu }}_{0},\dots ,\! {\varvec{\nu }}_{i}) \notin W_{i}'\) and \( ({\varvec{\nu }}_{0},\dots , {\varvec{\nu }}_{i-1}, {\varvec{\nu }}_{i}, {\varvec{0}}, \dots , {\varvec{0}}) \!\in \! {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}} \) lies in the neighborhood of \( ({\varvec{\nu }}_{0},\dots ,\! {\varvec{\nu }}_{i-1}, \!{\varvec{0}}, \!{\varvec{0}}, \!\dots ,\! {\varvec{0}})\) given by Proposition 2.11. Then, for \(k=0,\dots , s\) consider the family of convex piecewise affine functions \( \theta _{k,i}:\Delta _{i}\rightarrow {\mathbb {R}}\), \(i=0,\dots , s\), defined as \( \theta _{k,i}= \vartheta _{ {\varvec{\nu }}_{i}} \) if \(i<k\) and as \( \theta _{k,i}=0|_{\Delta _{i}}\) if \(i\ge k\). Their inf-convolutionis a convex piecewise affine function on \(\Delta \). By Corollary 2.10 and Proposition 2.11,is a tight incremental chain of mixed subdivisions of \(\Delta \). Moreover, Corollary 2.10 allows to choose the vector \(({\varvec{\nu }}_{0},\dots , {\varvec{\nu }}_{s-1})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}'}\) in the image of the linear map \(T_{{\varvec{{{\mathcal {A}}}}}'}\), so that the convex piecewise affine functions \(\theta _{k,i}\) are indeed affine.

The mixed volume of a family of polytopes \(\Delta _{i}\subset M_{{\mathbb {R}}}\), \(i=1,\dots , n\), is defined asFor \(n=0\), we agree that \( {\text {MV}}_{M}=1\).

The mixed integral of a family of convex piecewise affine functions \(\rho _i:\Delta _i\rightarrow {\mathbb {R}}\), \(i=0,\dots , n\), is defined as

For \(i=0,\dots , n\) let \(\rho _{i} :\Delta _{i}\rightarrow {\mathbb {R}}\) be a lattice convex piecewise affine function. Then, \({\text {MI}}_{M}(\rho _{0},\dots , \rho _{n})\in {\mathbb {Z}}\).

This follows directly from (2.17) and the analogous property for the mixed volume. \(\square \)

The sparse resultant, denoted by \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\), is defined as any primitive polynomial in \({\mathbb {Z}}[{\varvec{u}}]\) giving an equation for \(\varpi _{*}\Omega _{{\varvec{{{\mathcal {A}}}}}}\). The sparse eliminant, denoted by \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}}\), is defined as any irreducible polynomial in \({\mathbb {Z}}[{\varvec{u}}]\) giving an equation for \({\overline{\varpi (\Omega _{{\varvec{{{\mathcal {A}}}}}})}} \), if this a hypersurface, and as 1 otherwise.

Let \(\varphi :M\rightarrow M'\) be a monomorphism of lattices of rank n. Then, \({\text {Elim}}_{\varphi ({\varvec{{{\mathcal {A}}}}})}= {\pm } {\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}}\) and \({\text {Res}}_{\varphi ({\varvec{{{\mathcal {A}}}}})}= {\pm } {\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}^{[M':\varphi (M)]}\).

The monomorphism \(\varphi :M\rightarrow M'\) induces a finite map of degree \([M' : \varphi (M)]\)Setting \(F_{i}'=\sum _{a\in {{\mathcal {A}}}_{i}}u_{i,a} \, \chi ^{\varphi (a)}\) for the general Laurent polynomial with support \(\varphi ({{\mathcal {A}}}_{i})\) for each i, the system \(F'_0=\dots = F'_n=0\) has a nontrivial solution in \({\mathbb {T}}_{M'}\) if and only if \(F_0=\dots = F_n=0\) has a nontrivial solution in \({\mathbb {T}}_{M}\). Hence, \(\varphi ^{*}\) induces a commutative diagramwhich implies the stated equality between the sparse eliminants. From here, we also deduce that \({\overline{ \varpi ' (\Omega _{\varphi ({\varvec{{{\mathcal {A}}}}})})}} \) is not a hypersurface if and only if this also holds for \({\overline{\varpi (\Omega _{{\varvec{{{\mathcal {A}}}}}})}} \), in which case both \({\text {Res}}_{\varphi ({\varvec{{{\mathcal {A}}}}})}\) and \( {\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) are equal to \({\pm } 1\), proving the second equality in this case. Otherwise, the multiplicativity of the degree implies thatand so \(\varpi '_{*}\Omega _{\varphi ({\varvec{{{\mathcal {A}}}}})}= [M' : \varphi (M)] \,\varpi _{*}\Omega _{{\varvec{{{\mathcal {A}}}}}}\), which implies the second equality in this other case and completes the proof. \(\square \)

For \(i=0,\dots , n\) let \(f_{i}\in {\mathbb {C}}[M]\) with support contained in \({{\mathcal {A}}}_{i}\) and suppose that \({\text {Res}}_{{{\mathcal {A}}}_{1}^{v}, \dots , {{\mathcal {A}}}_{n}^{v}}({\text {init}}_{v}(f_{1}),\dots , {\text {init}}_{v}(f_{n})) \ne 0\) for all \(v\in N{\setminus }\{0\}\). Then,the first product being over the primitive vectors \(v\in N\) and the second over the solutions \(p\in {\mathbb {T}}_{M}\) of the system of equations \(f_{1}=\dots =f_{n}=0\), where \(m_{p}\) denotes the intersection multiplicity of this system of equations at the point p.

The fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}\) is the family of supports \({\varvec{{{\mathcal {A}}}}}_{J}\) for the minimal subset \(J\subset \{0,\dots , n\}\) such that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}} \in {\mathbb {Z}}[{\varvec{u}}_{J} ]\) or equivalently, such that \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}} \in {\mathbb {Z}}[{\varvec{u}}_{J} ]\).

By [33, Corollary 1.1 and Lemma 1.2] or [8, Proposition 3.13] when \(J\ne \emptyset \) the fundamental subfamily \({\varvec{{{\mathcal {A}}}}}_{J}\) coincides with the unique essential subfamily of \({\varvec{{{\mathcal {A}}}}}\), that is, the unique subfamily such that \({\text {rank}}( L_{{\varvec{{{\mathcal {A}}}}}_{J}}) = \#J-1\) and \( {\text {rank}}( L_{{\varvec{{{\mathcal {A}}}}}_{J'}}) \ge \#J'\) for all \(J'\subsetneq J\), whereas when \(J=\emptyset \), that is when \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}={\pm }1\), we have that \({\varvec{{{\mathcal {A}}}}}_{J}=\emptyset \) and \({\varvec{{{\mathcal {A}}}}}\) has at least two essential subfamilies.

Let \({\varvec{{{\mathcal {A}}}}}_{J}\) be the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}\) and suppose that \(J\ne \emptyset \). Then, \({\text {rank}}(L_{{\varvec{{{\mathcal {A}}}}}_{J}})= \#J - 1\), and the exponent in (3.4) can be written aswhere \(\pi \) is the projection \(M\rightarrow M/L_{{\varvec{{{\mathcal {A}}}}}_{J}}^{\mathrm{sat}}\).

The first claim follows from Remark 3.5, whereas the second is [8, Proposition 3.13]. \(\square \)

Let \({\varvec{{{\mathcal {A}}}}}_{J}\) be the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}\), suppose that \(J\ne \emptyset \) and consider \({\varvec{{{\mathcal {A}}}}}_{J}\) as a family of \(\#J\) nonempty finite subsets of the lattice \(L_{{\varvec{{{\mathcal {A}}}}}_{J}}\simeq {\mathbb {Z}}^{\#J-1}\). Then, \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}}={\pm }{\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}_{J}}={\pm }{\text {Res}}_{{\varvec{{{\mathcal {A}}}}}_{J}}\).

The first equality is given by [8, Proposition 3.11], whereas the second follows from the equality in (3.4) and Proposition 3.6. \(\square \)

Let \({\widetilde{{\varvec{{{\mathcal {A}}}}}}}=({\widetilde{{{\mathcal {A}}}}}_{0},\dots , {\widetilde{{{\mathcal {A}}}}}_{n})\) be a further family of supports in M such that \({\widetilde{{{\mathcal {A}}}}}_{i}\subset {{\mathcal {A}}}_{i}\) for all i. Let \({\widetilde{{\varvec{{{\mathcal {A}}}}}}}_{J}\) and \({\varvec{{{\mathcal {A}}}}}_{K}\) be the respective fundamental subfamilies of supports. Then, \(J\subset K\).

By the degree formula in (3.2), an index \(j\in \{0,\dots , n\}\) lies in K if and only ifThe statement follows then from the monotonicity of the mixed volume with respect to the inclusion of polytopes. \(\square \)

For \({\varvec{d}}=(d_{0},\dots , d_{n})\in ({\mathbb {N}}_{>0})^{n+1}\) let \({\text {Res}}_{{\varvec{d}}}\) be the homogeneous resultant, giving the condition for a system of \(n+1\) homogeneous polynomials in \(n+1\) variables of degrees \({\varvec{d}}\) to have a zero in the n-dimensional projective space [6, §3.2]. It coincides, up to the sign, both with the sparse eliminant and the sparse resultant for the lattice \(M={\mathbb {Z}}^{n}\) and the family of supports \({\varvec{{{\mathcal {A}}}}}=({{\mathcal {A}}}_{0},\dots , {{\mathcal {A}}}_{n})\) given bywhere for a lattice point \({\varvec{a}}=(a_{1},\dots , a_{n})\in {\mathbb {Z}}^{n}\) we denote by \(|{\varvec{a}}|=\sum _{i=1}^{n}a_{i}\) its length. Then, \(\Delta _{i}= \{{\varvec{x}}\in ({\mathbb {R}}_{\ge 0})^{n} \mid |{\varvec{x}}|\le d_{i}\}\) for each i and we can deduce from the degree formula in (3.2) that

Let \({\varvec{\omega }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) and let t be a variable. For \(P\in {\mathbb {C}}[{\varvec{u}}] {\setminus }\{0\}\) setThe order and the initial part of P with respect to \({\varvec{\omega }}\) are the elements \({\text {ord}}_{{\varvec{\omega }}}(P) \in {\mathbb {R}}\) and \({\text {init}}_{{\varvec{\omega }}}(P)\in {\mathbb {C}}[{\varvec{u}}]{\setminus }\{0\}\) defined by the equationwhere o(1) denotes a sum of terms whose degree in t is strictly positive.

For a nonzero rational function \(P\in {\mathbb {C}}({\varvec{u}})^{\times }\) written as \(P={P_{1}}/{P_{2}}\) with \(P_{i}\in {\mathbb {C}}[{\varvec{u}}] {\setminus }\{0\}\), \(i=1,2\), the order and the initial part of P with respect to \({\varvec{\omega }}\) are defined asThese notions do not depend on the choice of \(P_{1}\) and \(P_{2}\) and the mapsare group morphisms. We extend them by setting \({\text {ord}}_{{\varvec{\omega }}}(0)=+\infty \) and \({\text {init}}_{{\varvec{\omega }}}(0)=0\). The notion of initial part generalizes the definition in (3.3) for Laurent polynomials.

For \({\varvec{\omega }}=({\varvec{\omega }}_{0},\dots , {\varvec{\omega }}_{n})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) let \(\vartheta _{{\varvec{\omega }}_{i}}:\Delta _{i}\rightarrow {\mathbb {R}}\), \(i=0,\dots , n\), and \(\Theta _{{\varvec{\omega }}} :\Delta \rightarrow {\mathbb {R}}\) be the associated convex piecewise affine functions as in (2.8) and (2.9). Let D be an n-cell of the mixed subdivision \(S(\Theta _{{\varvec{\omega }}})\) of \(\Delta \) and \(D_{i} \in S(\vartheta _{{\varvec{\omega }}_{i}})\), \(i=0,\dots , n\), its components as defined in (2.7). The restriction of \({\varvec{{{\mathcal {A}}}}}\) to D is the family of nonempty finite subsets of M defined as

Let \({\varvec{\omega }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\). Then,

For \({\varvec{\omega }}=({\varvec{\omega }}_{0},\dots , {\varvec{\omega }}_{n})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) there is \({\widetilde{{\varvec{\omega }}}}=({\widetilde{{\varvec{\omega }}}}_{0},\dots , {\widetilde{{\varvec{\omega }}}}_{n})\in {\mathbb {Z}}^{{\varvec{{{\mathcal {A}}}}}}\) such that

Set \({{\mathcal {S}}}={\text {supp}}({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}) \subset {\mathbb {N}}^{{\varvec{{{\mathcal {A}}}}}}\) and \({{\mathcal {S}}}^{{\varvec{\omega }}}\) for the subset of \({{\mathcal {S}}}\) of lattice points with minimal scalar product with respect to \({\varvec{\omega }}\). A vector \({\widetilde{{\varvec{\omega }}}}=({\widetilde{{\varvec{\omega }}}}_{0},\dots , {\widetilde{{\varvec{\omega }}}}_{n})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) verifies the condition (1) if and only ifWith notation as in (2.3), for each \(D\in S(\Theta _{{\varvec{\omega }}})^{n}\) set \(v_{D}\in N_{{\mathbb {R}}}\) for the unique vector such that \( D=\Gamma (\Theta _{{\varvec{\omega }}},v_{D}) \). For each i let \(D_{i}^{0}\subset {{\mathcal {A}}}_{i}\) be the set of vertices of the i-th component of D. Then, \(D_{i}=\Gamma (\vartheta _{{\widetilde{{\varvec{\omega }}}}_{i}},v_{D})\) if and only ifIf this condition holds, then \(D=\Gamma (\Theta _{{\widetilde{{\varvec{\omega }}}}},v_{D})\) by Proposition 2.1(1).

Hence, if \({\widetilde{{\varvec{\omega }}}}\) satisfies the condition (3.7) and that in (3.8) for all \(D\in S(\Theta _{{\varvec{\omega }}})^{n}\), then it also verifies (1), (2) and (3). These conditions amount to the fact that \({\widetilde{{\varvec{\omega }}}}\) lies in the relative interior of a polyhedral cone of \({\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) defined over \({\mathbb {Z}}\). This relative interior is nonempty as it contains \({\varvec{\omega }}\), and so it also contains a vector in \( {\mathbb {Z}}^{{\varvec{{{\mathcal {A}}}}}}\). \(\square \)

Let \((v,l) \in N\times {\mathbb {Z}}\) be a primitive lattice vector with \(l>0\). Let \( (v,l)^{\perp } \) be its orthogonal subspace of \(M_{{\mathbb {R}}}\times {\mathbb {R}}\) andthe lattice map defined by \((a,q)\mapsto a\). Then, \([M : \varphi ((v,l)^{\perp } \cap (M\times {\mathbb {Z}}) )]=l.\)

Set for short \(P=(v,l)^{\perp } \cap (M\times {\mathbb {Z}})\), which is a sublattice of \(M\times {\mathbb {Z}}\) of rank n. The map \(\varphi :P\rightarrow M\) is injective if and only if so is its dual \(\varphi ^{\vee }:M^{\vee } \rightarrow P^{\vee }\) and if this is the case, thenWe have that \(M^{\vee }=N\) and \(P^{\vee }\simeq (M\times {\mathbb {Z}}) / {\mathbb {Z}}\, (v,l) \). With these identifications, the dual map \(\varphi ^{\vee }:N\rightarrow (M\times {\mathbb {Z}}) / {\mathbb {Z}}\, (v,l) \) writes down as \(\varphi ^{\vee }(w)= (w,0)+ {\mathbb {Z}}\, (v,l)\). Hence, \(\varphi ^{\vee }\) is injective because \(l>0 \). Moreover, its image is the sublattice \((M\times l\, {\mathbb {Z}}) / {\mathbb {Z}}\, (v,l) \) and sowhich together with (3.9) implies the statement. \(\square \)

By [32, Proposition 4.5], the degree of a monomial deformation of the sparse resultant can be computed in terms of mixed integrals asSince \({\text {ord}}_{{\varvec{\omega }}}({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}})= -\deg _{t}({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}^{-{\varvec{\omega }}})\), this gives the first part of the statement.

For the second part, we reduce without loss of generality to the case when \({\varvec{\omega }}\in {\mathbb {Z}}^{\varvec{{{\mathcal {A}}}}}\) thanks to Lemma 3.13. Set \({\varvec{F}}^{{\varvec{\omega }}}=(F_{0}^{{\varvec{\omega }}_{0}},\dots , F_{n}^{{\varvec{\omega }}_{n}})\) withWith notation as in (3.1) and (3.5), we have thatConsider the family of \(n+2\) nonempty finite subsets of \(M\times {\mathbb {Z}}\) given byLet \( {\varvec{v}}=\{v_{(0,0)}, v_{(0,1)} \}\) be a set of variables, so that the general Laurent polynomial with support \({{\mathcal {C}}}\) is \( v_{(0,0)}+v_{(0,1)} \, z \) and that with support \({\widehat{{{\mathcal {A}}}}}_{i}\) is \( F_{i}^{{\varvec{\omega }}}(z)\), the evaluation of \(F_{i}^{{\varvec{\omega }}}\) at \(t=z\) for each i. Set \({\widehat{{\varvec{{{\mathcal {A}}}}}}}= ({\widehat{{{\mathcal {A}}}}}_{0}, \dots , {\widehat{{{\mathcal {A}}}}}_{n})\). By the “hidden variable” formula in [8, Proposition 4.7], there is \(d_{{\varvec{\omega }}} \in {\mathbb {Z}}\) such thatThanks to the formula in (3.6), we have thatprovided this latter polynomial is nonzero. To see this, consider a family of Laurent polynomials \(f_{i}\in {\mathbb {C}}[M]\) with \({\text {supp}}(f_{i}) \subset {{\mathcal {A}}}_{i}\), \(i=0,\dots , n\), that is sufficiently generic and set \({\varvec{f}}^{{\varvec{\omega }}}=(f_{0}^{{\varvec{\omega }}_{0}},\dots , f_{n}^{{\varvec{\omega }}_{n}})\). By the invariance of the sparse resultant under translation of the first support and the Poisson formula (Theorem 3.3), with notation as therein we have thatthe products being over the primitive lattice vectors \((v,l)\in N\times {\mathbb {Z}}\), and where \({\widehat{{\varvec{{{\mathcal {A}}}}}}}^{(v,l)}\) denotes the family of supports \( ({\widehat{{{\mathcal {A}}}}}_{0}^{(v,l)}, \dots , {\widehat{{{\mathcal {A}}}}}_{n}^{(v,l)})\). The last equality follows from the fact that \(-h_{{{\mathcal {C}}}-(0,1)}((v,l))=\max \{0,l\}\). Since this holds for every choice of \({\varvec{f}}\), we have thatbecause \({\text {init}}_{(v,l)}({\varvec{F}}^{{\varvec{\omega }}}(z))\) is the general Laurent polynomial with support \({\widehat{{\varvec{{{\mathcal {A}}}}}}}^{(v,l)}\).

Let \((v,l)\in N\times \mathbb {Z}\) be a primitive vector such that \(l>0\). For the linear map \(\varphi :(v,l)^{\bot }\cap (M\times {\mathbb {Z}}) \rightarrow M\) induced from the projection onto the first factor we have that \([M:\varphi ((v,l)^{\bot }\cap (M\times {\mathbb {Z}}))]=l\) by Lemma 3.14. For the cell \(D=\Gamma \big (\Theta _{{\varvec{\omega }}},\frac{1}{l}\, v\big )\), we also have that \(\varphi ({\widehat{{{\mathcal {A}}}}}_{i}^{(v,l)})={{\mathcal {A}}}_{i}\cap D_{i}\) for each i. Proposition 3.2 then implies thatSince every n-cell of \(S(\Theta _{{\varvec{\omega }}})^{n}\) appears exactly once in the product (3.11), this second part then follows from (3.10), (3.11) and (3.12). \(\square \)

The function \(\mu _{{\varvec{\Delta }}}:N\rightarrow {\mathbb {Z}}\) given byis well defined and linear. Therefore \(\mu _{{\varvec{\Delta }}} \in M =N^{\vee } \).

Let \(v\in N\). For each subset \(I\subset \{0,\dots , n\}\) we have thatbecause v is linear. For \(v'\in N\), the definition of the mixed integral then implies thatwhich shows that \(\mu _{{\varvec{\Delta }}}\) is linear. Moreover, \(v|_{\Delta _{i}}\) is a lattice convex piecewise affine function and so Proposition 2.15 implies that \(\mu _{{\varvec{\Delta }}}(v)\in {\mathbb {Z}}\). The last claim follows from the previous ones. \(\square \)

The sparse resultant \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) is homogeneous with respect to \(\deg _{M}\) and

Let \(v\in N\). For the weight \({\varvec{\omega }}\in {\mathbb {Z}}^{{\varvec{{{\mathcal {A}}}}}}\) defined by \(\omega _{i,a}=\langle v,a\rangle \) for \(i=0, \dots , n \) and \(a\in {{\mathcal {A}}}_{i}\), we have that \(\vartheta _{{\varvec{\omega }}_{i}}=v|_{\Delta _{i}}\) for each i and \(\Theta _{{\varvec{\omega }}}=v|_{\Delta }\). Hence, \(\Delta \) is the unique n-cell of \(S(\Theta _{{\varvec{\omega }}})\), and from Theorem 3.12 we deduce thatThis implies that for all \({\varvec{c}}\in {\text {supp}}({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}})\) we have that \(\langle v, \sum _{i,a} c_{i,a}\, a \rangle = \langle v,\mu _{{\varvec{\Delta }}}\rangle =\mu _{{\varvec{\Delta }}}(v)\). Since this holds for all \(v\in N\), we deduce the statement. \(\square \)

When \(M={\mathbb {Z}}^{n}\) we have that \(\mu _{{\varvec{\Delta }}}= (\mu _{{\varvec{\Delta }},1},\dots , \mu _{{\varvec{\Delta }},n})\) withIn this case \({\mathbb {T}}_{M}=({\mathbb {C}}^{\times })^{n}\), and for a point \({\varvec{p}}=(p_{1},\dots , p_{n})\) in this torus we have that

Let \({\text {Res}}_{{\varvec{d}}}\) be the homogeneous resultant corresponding to a sequence of degrees \({\varvec{d}}=(d_{0},\dots , d_{n}) \in ({\mathbb {N}}_{>0})^{n+1}\) as in Example 3.9. Since the function \({\varvec{x}}\mapsto x_{i}\) is linear, for each i the mixed integral in (3.15) can be computed aswhere the last equality can be proven with elementary algebra as in [17, Theorem 3.7]. This gives the well-known isobarism of the homogeneous resultant, a result that goes back to Macaulay [28, page 11].

The following conditions are equivalent: If any of these conditions holds, then \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}_{D}} \in {\mathbb {Z}}[{\widetilde{{\varvec{u}}}}]\) for all \(D\in S(\Theta _{{\varvec{\omega }}})^{n}\) and

By Theorem 3.12, following Definition 3.10, we have thatthe product being over the n-cells D of \(S(\Theta _{{\varvec{\omega }}})\).

Since \({\varvec{\omega }}\in {\mathbb {N}}^{{\varvec{{{\mathcal {A}}}}}}\) we have that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}^{{\varvec{\omega }}} \in {\mathbb {C}}[{\varvec{u}}][t]\) and \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}^{{\varvec{\omega }}}\big |_{t=0}={\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}({\widetilde{{\varvec{F}}}})\). Hence, \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}({\widetilde{{\varvec{F}}}})\ne 0\) if and only if \({\text {ord}}_{{\varvec{\omega }}}({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}})=0\), and so the expression in (3.17) gives the equivalence between (1) and (2). If any of these conditions holds, thenSince the left-hand side of (3.18) lies in \({\mathbb {Z}}[{\widetilde{{\varvec{u}}}}]\) and the right-hand side is a polynomial, the factors of the latter lie in \({\mathbb {Z}}[{\widetilde{{\varvec{u}}}}]\), proving the last part of the statement and implying (3).

Conversely, suppose that the condition (3) holds. Evaluating the expression in (3.17) by setting \(u_{i,a}=0\) for \(i=0,\dots , n\) and \(a\in {{\mathcal {A}}}_{i}{\setminus } {\widetilde{{{\mathcal {A}}}}}_{i}\) we deduce thatwhich implies (1) and concludes the proof. \(\square \)

Let \(M={\mathbb {Z}}\), \({{\mathcal {A}}}_{0}={{\mathcal {A}}}_{1}=\{0,1\}\) and set \({\varvec{{{\mathcal {A}}}}}=({{\mathcal {A}}}_{0},{{\mathcal {A}}}_{1})\). Then,Set \({\widetilde{{{\mathcal {A}}}}}_{i}=\{0\}\), \(i=0,1\), and let \({\widetilde{{\varvec{F}}}}=(u_{0,0}, u_{1,0})\) be the corresponding system of Laurent polynomials in \({\mathbb {C}}[t^{{\pm }1}]\). With notation as in Theorem 3.19, in this case we have that \(\vartheta _{{\varvec{\omega }}_{i}}(x)= x\) for \(i=0,1\) and \(x\in [0,1]\) and so \(\Theta _{{\varvec{\omega }}}(x)=x\) for \(x\in [0,2]\), as shown in Fig. 2. Hence,This result then tells us that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}({\widetilde{{\varvec{F}}}})=0\), which can also be verified from (3.19).

Set also \({\widetilde{{{\mathcal {A}}}}}_{0}=\{0\}\) and \({\widetilde{{{\mathcal {A}}}}}_{1}=\{1\}\), and let \({\widetilde{{\varvec{F}}}}=(u_{0,0}, u_{1,1}\, t)\) be the corresponding system of Laurent polynomials. Then, \( \vartheta _{{\varvec{\omega }}_{0}}(x)= x\) and \(\vartheta _{{\varvec{\omega }}_{1}}(x)= 1-x\) for \(x\in [0,1]\), and so \(\Theta _{{\varvec{\omega }}}(x)= \max \{1-x, x-1\}\) for \(x\in [0,2]\), as shown in Fig. 3. Hence,and so Theorem 3.19 implies that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}({\widetilde{{\varvec{F}}}})\ne 0\). The mixed subdivision \(S(\Theta _{{\varvec{\omega }}})\) has the two 1-cells \(D=[0,1]\) and \(D'=[1,2]\), that decompose as \(D=0+[0,1]\) and \(D'=[0,1]+1\). Hence, this result also implies thatwhich can also be verified from (3.19).

The Minkowski sum \({\widetilde{\Delta }}= \sum _{i=0}^{n}{\widetilde{\Delta }}_{i}\) is the cell of the mixed subdivision \(S(\Theta _{{\varvec{\omega }}})\) corresponding to the vector \(0\in N_{{\mathbb {R}}}\) and its components are the polytopes \({\widetilde{\Delta }}_{i}\), \(i=0,\dots , n\). Hence, \({\varvec{{{\mathcal {A}}}}}_{{\widetilde{\Delta }}}=\varvec{\widetilde{\mathcal {A}}}\). We have that either \({\widetilde{\Delta }}\) is an n-cell of \(S(\Theta _{{\varvec{\omega }}})\) or \({\text {Res}}_{\varvec{\widetilde{\mathcal {A}}}}={\pm }1\). In the presence of any of the equivalent conditions in Theorem 3.19, the factorization in (3.16) holds and it can be alternatively written asWhen \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) is known, this factorization can be useful to compute the sparse resultant \({\text {Res}}_{\varvec{\widetilde{\mathcal {A}}}}\) as a factor of the evaluation \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}({\widetilde{{\varvec{F}}}})\).

Let \({\widetilde{{{\mathcal {A}}}}}_{0}\subset {{\mathcal {A}}}_{0}\) be a nonempty subset and \({\widetilde{F}}_{0}\) the general Laurent polynomial with support \({\widetilde{{{\mathcal {A}}}}}_{0}\). With notation as in Theorems 3.19 and 3.3, we have thatthe last product being over the primitive vectors \(v\in N\).

Let \({\widetilde{f}}_{0}\in {\mathbb {C}}[M]\) with \({\text {supp}}( {\widetilde{f}}_{0} ) \subset {\widetilde{{{\mathcal {A}}}}}_{0}\) and \(f_{i}\in {\mathbb {C}}[M]\) with \({\text {supp}}(f_{i})\subset {{\mathcal {A}}}_{i}\), \(i=1,\dots , n\), such that \({\text {Res}}_{{{\mathcal {A}}}_{1}^{v}, \dots , {{\mathcal {A}}}_{n}^{v}}({\text {init}}_{v}(f_{1}),\dots , {\text {init}}_{v}(f_{n})) \ne 0\) for all \(v\in N{\setminus }\{0\}\). By Theorem 3.3, we have thatthe first product being over the primitive vectors \(v\in N\) and the second over the solutions \(p\in {\mathbb {T}}_{M}\) of \(f_{1}=\dots = f_{n}=0\), where \(m_p\) denotes the corresponding intersection multiplicity, and similarlyTaking the quotient between these two formulae we deduce the second equality in the statement evaluated at \({\widetilde{f}}_{0}, f_{1}, \dots , f_{n}\). Since these Laurent polynomials are generic, we deduce that this equality holds for the general Laurent polynomials \({\widetilde{F}}_{0},F_{1}, \dots , F_{n}\), as stated. This also implies that \( {\text {Res}}_{{{\mathcal {A}}}_0,{{\mathcal {A}}}_{1},\ldots , {{\mathcal {A}}}_n}({\widetilde{F}}_0,F_1,\ldots , F_n) \ne 0\), and so the first equality follows from (3.16). \(\square \)

In [29], Minimair also studied the factorization of the evaluation of sparse resultant at systems of Laurent polynomials with smaller supports. Unfortunately, his result is not consistent, since its statement involves the exponent introduced in [29, Remark 3] that, as explained in [8, §5], is not well defined.

The row content function associated with \({\varvec{{{\mathcal {A}}}}}\), \({\varvec{\rho }}\) and \(\delta \) is the function \( {\text {rc}}:{{\mathcal {B}}}\rightarrow \bigcup _{i=0}^{n} (\{i\}\times {{\mathcal {A}}}_{i})\) defined by \({\text {rc}}(b)=(i(b),a(b))\) for \(b\in {{\mathcal {B}}}\).

Let \(i\in \{0,\dots , n\}\) and \(b\in {{\mathcal {B}}}_{i}\). Then, In particular \(\dim (V_{i})=\#{{\mathcal {B}}}_{i}\) and for all \(G\in V_{i}\) we have that \(G\, F_{i}\in V\).

Let \(b,b'\in {{\mathcal {B}}}_{i}\) such that \(b-a(b)=b'-a(b')\), and denote by C and \(C'\) the n-cells of \( S(\rho )\) corresponding to these lattice points. With notation as in  (2.6), the complementary cells \(C_{i}^{\mathrm{c}}\) and \(C_{i}^{\prime \mathrm{c}}\) have both dimension n and the lattice point \(b-a(b)=b'-a(b')\) lies both in \( {\text {ri}}(C_{i}^{\mathrm{c}})+\delta \) and in \( {\text {ri}}(C_{i}^{\prime \mathrm{c}})+\delta \), the translates of the relative interiors of these cells. This implies that \(C_{i}^{\mathrm{c}}=C_{i}^{\prime \mathrm{c}}\), and so \(C=C'\) by Proposition 2.2. We deduce that \(\{a(b)\}=C_{i}=C_{i}'=\{a(b')\} \) and so \(b=b'\), proving (1).

We also have that \( b-a(b)\in C_{i}^{\mathrm{c}} + \delta \subset \Delta _{i}^{\mathrm{c}}+\delta \) and soas stated in (2). The last two claims follow directly from (1) and (2). \(\square \)

The Sylvester map associated with \({\varvec{{{\mathcal {A}}}}}\), \({\varvec{\rho }}\) and \(\delta \) is the linear map \( \Phi _{{\varvec{{{\mathcal {A}}}}}, {\varvec{\rho }}, \delta }:\bigoplus _{i=0}^{n} V_{i} \rightarrow V\) given by the restriction of \(\Phi _{{\varvec{{{\mathcal {A}}}}}}\) to these linear subspaces. The Canny–Emiris matrix associated with \({\varvec{{{\mathcal {A}}}}}\), \({\varvec{\rho }}\) and \(\delta \), denoted by \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\delta } \in {\mathbb {K}}^{{{\mathcal {B}}}\times {{\mathcal {B}}}}\), is the matrix of this linear map in terms of row vectors. We set \( H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\delta }=\det ({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\delta }) \in {\mathbb {Z}}[{\varvec{u}}]\) for the corresponding Canny–Emiris determinant.

Since the vector \(\delta \) is fixed throughout our constructions, we omit it from the notation, and so this linear map, matrix and determinant will be, respectively, denoted by

For \({\varvec{G}}\in \bigoplus _{i}V_{i}\) we have that \([{\varvec{G}}] \cdot {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}} = [ \Phi _{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\varvec{G}})] \), where \([{\varvec{G}}] \) and \( [\Phi _{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\varvec{G}})] \) denote the row vectors of \({\varvec{G}}\) and of \( \Phi _{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\varvec{G}}) \) with respect to the bases of \( \bigoplus _{i} V_{i}\) and of V given by the decomposition in (4.3). Hence, the row of the Canny–Emiris matrix corresponding to an element \(b\in {{\mathcal {B}}}\) codifies the coefficients of the Laurent polynomial \(\chi ^{b-a(b)} \, F_{i(b)}\). Precisely, the entry corresponding to a pair \(b,b'\in {{\mathcal {B}}}\) is

The nonmixed index subset, denoted by \({{\mathcal {B}}}^{\circ } \), is the set of elements of \({{\mathcal {B}}}\) lying in the translated n-cells of \(S(\rho )\) that are not i-mixed for any i (Definition 2.3). We denote bythe corresponding principal submatrix and minor of \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\).

For \({{\mathcal {C}}}\subset {{\mathcal {B}}}\), the principal minor \(H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}, {{\mathcal {C}}}}\) is homogeneous in each set of variables \({\varvec{u}}_{i}\) and with respect to the grading \(\deg _{M}\) defined in (3.14). Moreover,

Let \(i\in \{0,\dots , n\}\). For \(b\in {{\mathcal {C}}}\), the entries in the corresponding row of \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}}\) are homogeneous in \({\varvec{u}}_{i}\) of degree 1 if \(i(b)=i\) and of degree 0 otherwise. Expanding \(H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}, {{\mathcal {C}}}}\) along rows, we deduce that it is homogeneous in \({\varvec{u}}_{i}\) of degree \(\#({{\mathcal {B}}}_{i}\cap {{\mathcal {C}}})\).

For the claims concerning \(\deg _{M}\), first extend this grading to \({\mathbb {C}}[{\varvec{u}}][M]\) by declaring that \(\deg _{M}(\chi ^{a})=a\) for \(a\in M\). Consider then the matrix \( {\widetilde{{{\mathcal {H}}}}}\in {\mathbb {C}}[{\varvec{u}}][M]^{{{\mathcal {C}}}\times {{\mathcal {C}}}}\) obtained from \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}}\) multiplying by \(\chi ^{b-a(b)}\) the row corresponding to a lattice point b, for each \(b\in {{\mathcal {C}}}\). By Remark 4.4, the entry corresponding to a pair \(b,b'\in {{\mathcal {C}}}\) isHence, for \(b'\in {{\mathcal {C}}}\), the entries in the corresponding column of \({\widetilde{{{\mathcal {H}}}}}\) are homogeneous with respect to \(\deg _{M}\) of degree \(b'\). Expanding the determinant \({\widetilde{H}}=\det ({\widetilde{{{\mathcal {H}}}}})\) along columns, we deduce that it is homogeneous with respect to \(\deg _{M}\) of degreeThese claims then follow from the fact that \( H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}, {{\mathcal {C}}}} = {\widetilde{H}} \cdot \prod _{b\in {{\mathcal {C}}}}\chi ^{-b+a(b)} \). \(\square \)

The argument for the homogeneity with respect to \(\deg _{M}\) of the principal minors of a Canny–Emiris matrix is an extension of that of Macaulay for the isobarism of the homogeneous resultant in [28, page 11].

Let D be an n-cell of \(S( \phi )\). Then,

Clearly, the restriction \(\rho _{i}|_{D_{i}}\) is a convex piecewise affine function on \(D_{i}\). Since \(S(\phi )\preceq S(\rho )\) we have that \(D_{i}\) is a union of n-cells of \(S(\rho _{i})\). Hence, \(\rho _{i}|_{D_{i}}\) is defined on the set of vertices of these n-cells and so on \({{\mathcal {A}}}_{i}\cap D_{i}\), which proves (1).

For (2), note that for \(x\in D\) we have that (respectively, \(\rho |_{D}(x)\)) is defined as the infimum of the sumwith \(x_{i} \in D_{i}\) (respectively, \(x_{i} \in \Delta _{i}\)) for all i such that \(\sum _{i=0}^{n}x_{i}=x\). Let \(C\in S(\rho )\) such that \(x\in C\) and \(C\subset D\). By Proposition 2.1(3), the infimum of the sum in (4.4) with \(x_{i}\in \Delta _{i}\), \(i=0,\dots , n\), such that \(\sum _{i=0}^{n}x_{i}=x\) is attained when \(x_{i}\in C_{i}\) for all i. Since \(S(\phi ) \preceq S(\rho )\), we have that \(C_{i}\subset D_{i}\) and so \(x_{i}\in D_{i}\) for all i. This implies that and so , as stated.

The statements in (3) and (4) follow directly from that in (2). \(\square \)

The matrix \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}}_{ D},{\varvec{\rho }}_{D},{{\mathcal {C}}}_{D}}\) is the evaluation of the principal submatrix \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}}, {\varvec{\rho }}, {{\mathcal {C}}}_{D}}\) by setting \(u_{i,a} =0\) for \(i=0,\dots , n\) and \(a\in {{\mathcal {A}}}_{i}{\setminus } D_{i}\).

By Proposition 4.8, the row content function associated with the restricted data \(({\varvec{{{\mathcal {A}}}}}_{D}, {\varvec{\rho }}_{D}, \delta )\) coincides with that of \(({\varvec{{{\mathcal {A}}}}}, {\varvec{\rho }}, \delta )\) restricted to the index set \({{\mathcal {B}}}_{D}\).

For each i, the general Laurent polynomial \(F_{D,i}\) is the evaluation of \(F_{i}\) setting \(u_{i,a} =0\) for \(a\in {{\mathcal {A}}}_{i}{\setminus } D_{i}\). Hence, the Sylvester map \(\Phi _{{\varvec{{{\mathcal {A}}}}}_{ D},{\varvec{\rho }}_{D}}\) is the restriction of \(\Phi _{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) to the linear subspace \(\bigoplus _{i=0}^{n} V_{D,i}\) composed with the evaluation that sets \(u_{i,a} =0\) for all i and \(a\in {{\mathcal {A}}}_{i}{\setminus } D_{i}\).

This implies the statement when \({{\mathcal {C}}}= {{\mathcal {B}}}\). The case of an arbitrary subset \({{\mathcal {C}}}\subset {{\mathcal {B}}}\) follows from this one by considering the corresponding principal submatrices. \(\square \)

Set \({\varvec{\omega }}=(\phi _{i}(a))_{i,a}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) and let \({{\mathcal {C}}}\subset {{\mathcal {B}}}\). Then,

Let \(b , b'\in {{\mathcal {B}}}\) such that \(b'\in b-a(b)+{{\mathcal {A}}}_{i(b)}\) and set \(a'=b'-b+a(b)\in {{\mathcal {A}}}_{i(b)}\). Then,and the equality holds if and only if there is \(D\in S(\phi )^{n} \) with \(b,b'\in D+\delta \) and \(a'\in D_{i(b)}\).

With notation as in (2.6), we have that \(b-\delta -a(b) \in \Delta _{i(b)}^{\mathrm{c}}\) and \(a'\in \Delta _{i(b)}\). Since \(\phi =\phi _{i(b)}^{\mathrm{c}}\boxplus \phi _{i(b)}\), this implies thatLet \(C\in S(\rho )^{n}\) such that \(b\in C+\delta \) and \(D\in S(\phi )^{n}\) with \(C\subset D\). Then, \(b-a(b)-\delta \in C_{i(b)}^{\mathrm{c}} \) and \(a(b)\in C_{i(b)}\). Since \(S(\rho )\succeq S(\phi )\), we have that \( C_{i(b)}^{\mathrm{c}} \subset D_{i(b)}^{\mathrm{c}}\) and \( C_{i(b)} \subset D_{i(b)}\) and soProposition 2.1(3) then implies that \(\phi (b-\delta )= \phi _{i(b)}^{\mathrm{c}} (b-\delta -a(b))+\phi _{i(b)}(a(b))\). The inequality in (4.7) follows from this together with (4.8).

Now if \(b'\in D+\delta \) and \(a'\in D_{i(b)}\) then Proposition 2.1(3) together with (4.9) implies that the inequality in (4.8) is an equality, and so is (4.7).

Conversely suppose that (4.7) is an equality or equivalently, that this is the case for (4.8). Let \(D'\in S(\phi )^{n}\) such that \(b'\in D'+\delta \). Applying again Proposition 2.1(3),Since \(S(\rho )\) is tight we have that \(\dim (C_{i(b)}^{\mathrm{c}})=n-\dim (C_{i(b)})=n\), and since \(b-\delta \in {\text {ri}}(C)\) we also have that \( b-\delta -a(b) \in {\text {ri}}(C_{i(b)}^{\mathrm{c}})\). Hence, \( {\text {ri}}(C_{i(b)}^{\mathrm{c}}) \subset {\text {ri}}(D^{\prime \mathrm{c}}_{i(b)})\) and so \( b-\delta -a(b) \in {\text {ri}}(D_{i(b)}^{\prime \mathrm{c}})\). Using (4.9) we deduce that the n-cells \(D_{i(b)}^{\prime \mathrm{c}}\) and \(D_{i(b)}^{\mathrm{c}}\) coincide. Proposition 2.2 then implies that \(D^{\prime }=D\), completing the proof. \(\square \)

Let \(b , b'\in {{\mathcal {B}}}\) such that \(b'\in b-a(b)+{{\mathcal {A}}}_{i(b)}\) and set \(a'=b'-b+a(b)\in {{\mathcal {A}}}_{i(b)}\). Then,and the equality holds if and only if \(b'=b\).

Let \({\varvec{\omega }}=({\varvec{\omega }}_{0}, \dots , {\varvec{\omega }}_{n})\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) such that \(\vartheta _{{\varvec{\omega }}_{i}}=\rho _{i}\) for all i and \({{\mathcal {C}}}\subset {{\mathcal {B}}}\). Then,In particular \(H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}, {{\mathcal {C}}}}\ne 0\).

Set \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}}^{{\varvec{\omega }}}= {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}}( (t^{\omega _{i,a}}\, u_{i,a} )_{i\in \{0,\dots , n\}, a\in {{\mathcal {A}}}_{i}}) \in {\mathbb {K}}(t)^{{{\mathcal {C}}}\times {{\mathcal {C}}}}\) and let \({\widetilde{{{\mathcal {H}}}}}\) be the matrix obtained from it multiplying by \(t^{\rho (b-\delta ) - \rho _{i(b)}(a(b))}\) the row corresponding to a lattice point b, for each \(b\in {{\mathcal {C}}}\). The entry corresponding to a pair \(b,b'\in {{\mathcal {C}}}\) iswith \(a'=b'-b+a(b)\). For \(b'\in {{\mathcal {C}}}\) we have that \( \rho _{i(b)}(a') \le \omega _{i(b),a'}\) by the definition of this piecewise affine function. Moreover, let \(C \in S(\rho )^{n}\) such that \(b\in C+\delta \). Then, \(C_{i(b)}=\{a(b)\}\), and since \(\rho _{i(b)}= \vartheta _{{\varvec{\omega }}_{i(b)}}\) this implies that \(\rho _{i(b)}(a(b))= \omega _{i(b),a(b)}\). By Corollary 4.12and the equality holds if and only if \(b'=b\). Hence, for \(b'\in {{\mathcal {B}}}\) the entry in the corresponding column of \({\widetilde{{{\mathcal {H}}}}}\) for \(b\in {{\mathcal {C}}}\) is of order at least \(\rho (b'-\delta )\), and this value is only attained when \(b=b'\). We have that \( {\widetilde{{{\mathcal {H}}}}}[b,b]= u_{i(b),a(b) } \,t^{\rho (b-\delta )}\) and soproving the statement. \(\square \)

This result can be proven similarly as it was done for Proposition 4.13, by considering the matrix \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}}^{{\varvec{\omega }}}= {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}} ( (t^{\omega _{i,a}}\, u_{i,a} )_{ i, a}) \in {\mathbb {K}}(t)^{{{\mathcal {C}}}\times {{\mathcal {C}}}}\) and the modified matrix \({\widetilde{{{\mathcal {H}}}}}\) obtained multiplying by \(t^{\phi (b-\delta ) - \phi _{i(b)}(a(b))}\) the row of \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},{{\mathcal {C}}}}^{{\varvec{\omega }}}\) corresponding to a lattice point b, for each \(b\in {{\mathcal {C}}}\).

Let \(D\in S(\phi )^{n}\). By Lemma 4.11, the lowest order in t in the column of \({\widetilde{{{\mathcal {H}}}}}\) corresponding to a lattice point \(b' \in {{\mathcal {C}}}\cap (D+\delta ) \) is \(\phi (b'-\delta )\), and it is attained exactly when \(b\in D+\delta \) and \(a'=b'-b+a(b) \in D_{i(b)}\). Hence, the matrix extracted from \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}, {{\mathcal {C}}}}\) by keeping only these entries of minimal order in each column is block diagonal, with blocks corresponding to the n-cells of \(S(\phi )\). Moreover, the block corresponding to an n-cell D coincides with \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}}_{D},{\varvec{\rho }}_{D}, {{\mathcal {C}}}_{D}}\). Hence,By Proposition 4.13, all the \(H_{{\varvec{{{\mathcal {A}}}}}_{D},{\varvec{\rho }}_{D}, {{\mathcal {C}}}_{D}}\)’s are nonzero, which completes the proof. \(\square \)

\({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}} \mid H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) in \({\mathbb {Z}}[{\varvec{u}}]\).

For \(p\in {\mathbb {T}}_{M}\) we have that \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\cdot \zeta _{p}^{\mathrm{T}} = \sum _{i=0}^{n} F_{i}(p) \, \eta _{p,i}^{\mathrm{T}}\).

In terms of the dual basis of V, right multiplication of a row vector by \(\zeta _{p}^{\mathrm{T}}\) corresponds to the linear functional \({\text {eval}}_{p}:V \rightarrow {\mathbb {R}}\) defined by \(G\mapsto G(p)\). In terms of the dual basis of \(\bigoplus _{j}V_{j}\), right multiplication by \(\eta _{p,i}^{\mathrm{T}}\) corresponds to the linear functional \({\text {eval}}_{p,i}:\bigoplus _{j }V_{j} \rightarrow {\mathbb {R}}\) defined by \((G_{0},\dots , G_{n})\mapsto G_{i}(p)\). With these identifications, for \({\varvec{G}}\in \bigoplus _{j }V_{j}\) we have thatThe lemma follows from the fact that this equality is valid for every \({\varvec{G}}\). \(\square \)

If \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}}={\pm }1\) then the statement is trivial. Else, by Definition 3.1 we have that \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}} \in {\mathbb {Z}}[{\varvec{u}}]\) is irreducible and the points \({\varvec{u}}=({\varvec{u}}_{0},\dots , {\varvec{u}}_{n})\in {\mathbb {C}}^{{\varvec{{{\mathcal {A}}}}}}\) such that there exists \(p\in {\mathbb {T}}_{M}({\mathbb {C}})\) with \(F_{0}({\varvec{u}}_{0},p)=\dots = F_{n}({\varvec{u}}_{n},p)=0 \) form a dense subset of the hypersurface \(Z({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}})\subset \prod _{i}{\mathbb {P}}({\mathbb {C}}^{{{\mathcal {A}}}_{i}})\). By Lemma 4.15, for all these points we have that \( {{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\varvec{u}})\cdot \zeta _{p}^{\mathrm{T}}=0 \) and so \(\ker ({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\varvec{u}}))\ne 0\). Hence, \(H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\varvec{u}})=0\), which implies that \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}} \mid H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) in \({\mathbb {Z}}[{\varvec{u}}]\), as stated. \(\square \)

Let \(i\in \{0,\dots , n\}\) such that \({{\mathcal {B}}}_{i}\) is contained in the union of the translated i-mixed n-cells of \(S(\rho )\). Then,Moreover, if \({{\mathcal {B}}}_{i}\ne \emptyset \) then \({H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}}/{{\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}} \in {\mathbb {Z}}[{\varvec{u}}_{0},\dots , {\varvec{u}}_{i-1},{\varvec{u}}_{i+1},\dots , {\varvec{u}}_{n}]\).

For each i, the function \({{\mathcal {B}}}_{i}\rightarrow M\) defined by \( b\mapsto b-a(b)\) gives a bijection between The cardinality of both sets is equal to \(m_{i}\).

Denote by \({{\mathcal {C}}}_{i}\) and \({{\mathcal {C}}}_{i}'\) the finite subsets of M defined in (1) and in (2), respectively. For \(b\in {{\mathcal {C}}}_{i}\) let C be an i-mixed n-cell of \(S(\rho )\) with \(b\in C+\delta \). Then, \(C_{i}^{\mathrm{c}} \) is a mixed n-cell of \(S(\rho _{i}^{\mathrm{c}})\) and \(b-a(b)\in C_{i}^{\mathrm{c}}+\delta \subset \Delta _{i}^{\mathrm{c}}+\delta \), and so \( b-a(b) \in {{\mathcal {C}}}_{i}'\). Hence, the assignment \( b\mapsto b-a(b)\) defines a function \({{\mathcal {C}}}_{i}\rightarrow {{\mathcal {C}}}_{i}'\) which, by Lemma 4.2(1), is injective.

Now for \(c\in {{\mathcal {C}}}_{i}'\) let B be a mixed n-cell of \(S(\rho _{i}^{\mathrm{c}})\) with \(c\in B+\delta \). With notation as in (2.3), let \(v\in N_{{\mathbb {R}}}\) be the unique vector such that \( B=\Gamma (\rho _{i}^{\mathrm{c}}, v)\) and setBy Proposition 2.1(1), C is an i-mixed n-cell of \(S(\rho )\) and \(C_{i}^{\mathrm{c}}=B\) and moreover, \(C_{i}\) consists of a single lattice point \(a\in {{\mathcal {A}}}_{i}\). Setting \(b=c+a \in C+\delta \), we have that \(b-a(b)=b-a=c\) and so \({{\mathcal {C}}}_{i}\rightarrow {{\mathcal {C}}}_{i}'\) is surjective, proving the first claim.

For the second, note that each mixed n-cell B of \( S(\rho _{i}^{\mathrm{c}})\) is a lattice parallelepiped, and so the genericity condition in (4.1) implies that \(\# (B+\delta )\cap M= {\text {vol}}_{M}(B)\). Hence, the cardinality of \({{\mathcal {C}}}_{i}'\) is equal to the sum of the volumes of these n-cells which, by the formula in (2.16), coincides with \(m_{i}\). \(\square \)

For \(i=0,\dots , n\) let \({{\mathcal {C}}}_{i}\) be the set of lattice points in \({{\mathcal {B}}}_{i}\) lying in translated i-mixed n-cells of \(S(\rho )\). Then, there is a proper algebraic subset \(Y_{i}\subset \prod _{j\ne i} {\mathbb {C}}^{{{\mathcal {A}}}_{j}}\) such that for \((\overline{{\varvec{u}}}_{j})_{j\ne i}\in \prod _{j\ne i} {\mathbb {C}}^{{{\mathcal {A}}}_{j}} {\setminus } Y_{i}\), the zero sethas cardinality \(m_{i}\) and the matrix \( (\chi ^{b-a(b)}(p))_{b\in {{\mathcal {C}}}_{i}, p\in Z_{i}} \in {\mathbb {C}}^{m_{i}\times m_{i}}\) is nonsingular.

Suppose without loss of generality that \(i=0\). By Bernstein’s theorem [2] and the Bertini-type theorem in [23, Part I, Theorem 6.3(3)], there is a proper algebraic subset \(Y_{0}\subset \prod _{j=1}^{n} {\mathbb {C}}^{{{\mathcal {A}}}_{j}}\) such that for \(({\overline{{\varvec{u}}}}_{1}, \dots , {\overline{{\varvec{u}}}}_{n})\in \prod _{j=1}^{n} {\mathbb {C}}^{{{\mathcal {A}}}_{j}}{\setminus } Y_{0}\), if we set \(f_{j}=F_{j}(\overline{{\varvec{u}}}_{j}, \cdot )\in {\mathbb {C}}[M]\), \(j=0,\dots , n\), and \(Z_{0}=Z(f_{1},\dots , f_{n})\), then the ideal \((f_{1},\dots , f_{n})\subset {\mathbb {C}}[M]\) is radical and \(Z_{0}\) has cardinality \(m_{0}\).

By Lemma 4.17, the set \(\{b-a(b)\}_{b\in {{\mathcal {C}}}_{0}}\) coincides with the set of lattice points in \(\Delta _{0}^{\mathrm{c}}+\delta =(\sum _{j=1}^{n}\Delta _{j}) +\delta \) lying in translated mixed n-cells of \(S(\rho _{0}^{\mathrm{c}})\). After possibly enlarging \(Y_{0}\), by [30, Theorem 1.1] or [14, Theorem 4.1] the monomials \(\chi ^{b-a(b)}\), \(b\in {{\mathcal {C}}}_{0}\), form a basis of the quotient algebra \({\mathbb {C}}[M]/(f_{1},\dots , f_{n})\).

Since the ideal \((f_{1},\dots , f_{n})\) is radical, the map \( {\mathbb {C}}[M]/(f_{1},\dots , f_{n}) \rightarrow {\mathbb {C}}^{Z_{0}}\) defined by \(g\mapsto (g(p))_{p\in Z_{0}}\) is an isomorphism. Hence, the vectors \((\chi ^{b-a(b)}(p))_{p\in Z_{0}} \in {\mathbb {C}}^{Z_{0}}\), \(b\in {{\mathcal {C}}}_{0}\), are linearly independent, proving the lemma. \(\square \)

By its definition in (4.2), the set \({{\mathcal {B}}}_{i}\) contains the set of lattice points in the translated i-mixed n-cells of \(S(\rho )\). Thus, the hypothesis in the present statement amounts to the fact that \({{\mathcal {B}}}_{i}\) is equal to this set of lattice points. Lemma 4.17, Proposition 4.6 and the degree formula in (3.2) then imply thatWhen \(m_{i}=0\) the statement is clear. Hence, we suppose that \(m_{i}>0\). Consider then the \(2 \times 2\)-block decompositionwhere the first rows and the first columns correspond to \({{\mathcal {B}}}_{i}\) and the others to \({{\mathcal {B}}}_{j}\) for \(j\ne i\). By Proposition 4.13, the matrix \({{\mathcal {H}}}_{2,2}\) is nonsingular and sowith \({{\mathcal {H}}}'={{\mathcal {H}}}_{1,1}- {{\mathcal {H}}}_{1,2}\cdot {{\mathcal {H}}}_{2,2}^{-1}\cdot {{\mathcal {H}}}_{2,1}\).

With notation as in Lemma 4.18, choose \((\overline{{\varvec{u}}}_{j})_{j\ne i}\in \prod _{j\ne i} {\mathbb {C}}^{{{\mathcal {A}}}_{j}} {\setminus } Y_{i}\). For \({\overline{{\varvec{u}}}}_{i}\in {\mathbb {C}}^{{{\mathcal {A}}}_{i}}\) set \(\overline{{\varvec{u}}}=({\overline{{\varvec{u}}}}_{0},\dots , {\overline{{\varvec{u}}}}_{n})\in {\mathbb {C}}^{{\varvec{{{\mathcal {A}}}}}}\). Set also \(f_{j}=F_{j}({\overline{{\varvec{u}}}}_{j},\cdot )\in {\mathbb {C}}[M]\) for each j and denote by \(Z_{i}\subset {\mathbb {T}}_{M}\) the zero set of the Laurent polynomials \(f_{j}\), \(j\ne i\). With notation as in (4.10), for each \(p\in Z_{i}\) we have thatthanks to Lemma 4.15. Consider the matrices in \( {\mathbb {C}}^{{{\mathcal {B}}}_{i}\times Z_{i}} \simeq {\mathbb {C}}^{m_{i}\times m_{i}}\) defined asFrom (4.12) and (4.13), we deduce that \( {{\mathcal {H}}}'({\overline{{\varvec{u}}}})\cdot {{\mathcal {P}}}= {\text {diag}}((f_{i}(p))_{p\in Z_{i}}) \cdot {{\mathcal {Q}}}\) and soBy Lemma 4.18, the matrix \({{\mathcal {Q}}}\) is nonsingular and so \( H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}({\overline{{\varvec{u}}}})=0\) only if there is \(p\in Z_{i}\) such that \(f_{i}(p)=0\). Hence, both \(H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) and \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) are polynomials of degree \(m_{i}>0\) in the set of variables \({\varvec{u}}_{i}\) that, for a generic choice of \((\overline{{\varvec{u}}}_{j})_{j\ne i}\in \prod _{j\ne i} {\mathbb {C}}^{{{\mathcal {A}}}_{j}} \) vanish for \({\overline{{\varvec{u}}}}_{i}\in {\mathbb {C}}^{{{\mathcal {A}}}_{i}}\) if and only if this also holds for the irreducible polynomial \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}}\) (recall that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) is a power of \({\text {Elim}}_{{\varvec{{{\mathcal {A}}}}}}\)). Thus,with \(\gamma \in {\mathbb {Q}}({\varvec{u}}_{0},\dots , {\varvec{u}}_{i-1},{\varvec{u}}_{i+1},\dots , {\varvec{u}}_{n})^{\times }\), proving the first claim. The second is a direct consequence of the first together with Gauss’ lemma, since \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) is a primitive polynomial in \({\mathbb {Z}}[{\varvec{u}}_{0},\dots , {\varvec{u}}_{i-1},{\varvec{u}}_{i+1},\dots , {\varvec{u}}_{n}][{\varvec{u}}_{i}]\). \(\square \)

We have that \({H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}}/{{\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}} \in {\mathbb {Q}}({\varvec{u}}_{1},\dots , {\varvec{u}}_{n})\). Moreover, if \(m_{0}>0\) then \({H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}}/{{\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}} \in {\mathbb {Z}}[{\varvec{u}}_{1},\dots , {\varvec{u}}_{n}]\).

For \(b\in {{\mathcal {B}}}_{0}\) let C be the n-cell of \(S(\rho )\) such that \(b\in C+\delta \). Then, \(\dim (C_{i})>0\) for all \(i>0\) and so C is 0-mixed. Moreover, if \(m_{0}>0\) then \({{\mathcal {B}}}_{0}\ne \emptyset \) thanks to Lemma 4.17. The corollary follows then from Proposition 4.16. \(\square \)

We conjecture that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\mid H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) in the general case. If this is true, then the hypothesis \({{\mathcal {B}}}_{i}\ne \emptyset \) in Proposition 4.16 and that \(m_{0}>0\) in Corollary 4.19 would not be necessary.

Suppose that the fundamental subfamily of supports coincides with \({\varvec{{{\mathcal {A}}}}}\) and choose a permutation \(\sigma _{i}\) of the index set \(\{0,\dots , n\}\) with \(\sigma _{i}(0)=i\) for each i. Then,

Corollary 4.19 applied to the data \((\sigma _{i}({\varvec{{{\mathcal {A}}}}}), \sigma _{i}({\varvec{\rho }}), \delta )\) implies that \( {\text {Res}}_{\sigma _{i}({\varvec{{{\mathcal {A}}}}})}\) divides \( H_{\sigma _{i}({\varvec{{{\mathcal {A}}}}}), \sigma _{i}({\varvec{\rho }})}\) and that both polynomials have the same degree in the set of variables \({\varvec{u}}_{0}\). By the invariance of the sparse resultant under permutations of the supports we deduce that \( {\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}({\varvec{u}}) \) divides \( H_{\sigma _{i}({\varvec{{{\mathcal {A}}}}}), \sigma _{i}({\varvec{\rho }})} (\sigma _{i}({\varvec{u}})) \) and that both polynomials have the same degree in the set of variables \({\varvec{u}}_{i}\) for each i, which implies the statement. \(\square \)

The tight mixed subdivision \(S(\rho )\) is admissible if there is an incremental chain of mixed subdivisions of \(\Delta \) (Definition 2.4)with \(S(\theta _{n})\preceq S(\rho )\) such that for \(k=0,\dots , n\), each n-cell D of \(S(\theta _{k})\) verifies at least one of the conditions: The incremental chain of mixed subdivisions in (4.14) is called admissible (for \(S(\rho )\)).

Let \(S(\theta _{0})\preceq \dots \preceq S(\theta _{n})\) be an incremental chain of mixed subdivisions of \(\Delta \) such that for \(k=0,\dots , n\), each n-cell D of \(S(\theta _{k})\) verifies at least one of the conditions: Then, this incremental chain is admissible for any tight mixed subdivision of \(\Delta \) that refines \(S(\theta _{n})\).

Let \(k=0,\dots , n\) and \(D \in S(\theta _{k})^{n}\). If this n-cell satisfies the condition (1), then for \(i=0,\dots , n\) we have thatBy the basic properties of the mixed volume, \({\text {MV}}_{M}(D_{0},\dots , D_{i-1},D_{i+1},\dots , D_{n})=0\). Hence, \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}_{D}}={\pm }1\) thanks to the degree formula in (3.2) and so the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}_{D}\) is empty. Thus, in this case D satisfies the condition (1) in Definition 4.22.

If D satisfies the condition (2), then also \({\text {MV}}_{M}(D_{0},\dots , D_{j-1}, D_{j+1}, \dots , D_{n})=0\) for all \(j\ne i \) and so the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}_{D}\) is either empty or consists of the single support \({{\mathcal {A}}}_{i}\). In both cases, D also satisfies the condition (1) in Definition 4.22.

Finally suppose that D satisfies the condition (3). For \(b\in {{\mathcal {B}}}_{D,k}\) let C be the n-cell of \(S(\rho _{D})\) such that \(b\in C+\delta \). We have that \(\dim (C_{k})=0\) and that \(C_{j}\subset D_{j}\) for all j, and so for each \(i<k\) we have thatSince \(b\in {{\mathcal {B}}}_{D,k}\) we also have that \(\dim (C_{i}) >0\) for all \(i>k\) and so C is k-mixed. Hence, in this case D satisfies the condition (2) in Definition 4.22. We conclude that the incremental chain is admissible for any tight mixed subdivision of \(\Delta \) refining \(S(\theta _{n})\). \(\square \)

An incremental chain \(S(\theta _{0})\preceq \dots \preceq S(\theta _{n})\) of mixed subdivisions of \(\Delta \) with \(S(\theta _{n})\preceq S(\rho )\) that is tight in the sense of Definition 2.4, satisfies the conditions in Proposition 4.23. In particular, it is admissible.

Since the incremental chain is tight, for \(k=0,\dots , n\) and \(D \in S(\theta _{k})^{n}\) we have thatIf there is \(i<k\) such that \(D_{i}\) is a point, then D satisfies the condition (2) in Proposition 4.23. Else \(\dim (D_i)>0\) for all \(i< k\) and so the equality in (4.15) implies thatand so D satisfies the condition (3) in Proposition 4.23. This gives the first claim, whereas the second is an application of that proposition. \(\square \)

The incremental chains of mixed subdivisions \( S(\theta _{0})\preceq \dots \preceq S(\theta _{n})\) of \(\Delta \) obtained by setting \(s=n\) in Example 2.12, are admissible for \(S(\theta _{n})\).

Let \({\varvec{{{\mathcal {A}}}}}_{J}\) be the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}\). Then,

The first claim in (1) is a direct consequence of the hypothesis that \(J=\emptyset \). The same hypothesis together with the degree formula in (3.2) implies that \(m_{j} =0\) for all j. Lemma 4.17 then implies that \({{\mathcal {B}}}^{\circ }={{\mathcal {B}}}\), which gives the second claim.

For (2), first note that, by the rank condition in Proposition 3.6, if \(\#J =1\) for the fundamental subfamily \({\varvec{{{\mathcal {A}}}}}_J\), its unique element \({{\mathcal {A}}}_i\) is a singleton. Then, in this case, \({\varvec{u}}_{i }=\{u_{i ,a }\}\) and \(m_{j}=0\) for all \(j\ne i \). The first claim follows then from the fact that \({\text {Res}}_{{\varvec{{{\mathcal {A}}}}}}\) is a primitive homogeneous polynomial in \({\mathbb {Z}}[u_{i ,a }]\) of degree \(m_{i }\). The hypothesis that \({{\mathcal {A}}}_{i }=\{a \}\) also implies that \(S(\rho )\) has no n-cells that are j-mixed for \(j\ne i \). By Lemma 4.17, the subset \({{\mathcal {C}}}_{i} \subset {{\mathcal {B}}}_{i }\) of lattice points lying in the translated n-cells that are i-mixed has cardinality \(m_{i }\). For each \(b \in {{\mathcal {C}}}_{i}\) we have that \({\text {rc}}(b)=(i ,a )\) and so, for \(b'\in {{\mathcal {B}}}\),Since \({{\mathcal {B}}}^{\circ }={{\mathcal {B}}}{\setminus } {{\mathcal {C}}}_{i}\) and \({{\mathcal {E}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) is the principal submatrix of \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) corresponding to this subset, we deduce the second claim. \(\square \)

Suppose that \(S(\rho )\) is admissible and let \(S(\theta _{0})\preceq \dots \preceq S(\theta _{n})\) with \(S(\theta _{n})\preceq S(\rho )\) be an admissible incremental chain for it. Then,the product in the second formula being over the n-cells D of \(S(\theta _{1})\).

Let \(S(\theta _{0})\preceq \dots \preceq S(\theta _{n})\) with \(S(\theta _{n})\preceq S(\rho )\) be an admissible incremental chain, and for \(k=0,\dots , n\) let \(\theta _{k,j}:\Delta _{j}\rightarrow {\mathbb {R}}\), \(j=0,\dots , n\), be the family of convex piecewise affine functions corresponding to \(\theta _{k}\). Set also \(\theta _{n+1}=\rho \).

We prove by reverse induction on k that for every n-cell D of \( S(\theta _{k})\) we have thatThe first statement in the theorem corresponds to the case when \(k=0\) and \(D=\Delta \).

For \(k=n+1\) we note that \(S(\theta _{n+1})=S(\rho )\) is tight. Hence, for \(D\in S(\theta _{n+1})\) we have that \(\sum _{i=0}^{n}\dim (D_{i})=n\) and so \(\dim (D_{i})=0\) for at least one i. This implies that the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}_{D}\) is either empty or consists of the single support \({{\mathcal {A}}}_{i} \cap D_{i}\). Hence, \({\varvec{{{\mathcal {A}}}}}_{D}\) verifies the hypothesis of Proposition 4.26, and so the equality in (4.16) follows from this result.

Hence, suppose that \(0\le k\le n\) and let \(D\in S(\theta _{k})\). In case D satisfies the condition (1) in Definition 4.22, the equality in (4.16) follows similarly from Proposition 4.26. If this does not happen, then D satisfies the condition (2) in this definition, and so the subset \({{\mathcal {B}}}_{D,k}\) is contained in the union of the translated k-mixed n-cells of \(S(\rho _{D})\).

By Proposition 3.8, for all \(i<k\) and every n-cell C of \(S(\theta _{i})\) containing D, the index set of the fundamental subfamily of \({\varvec{{{\mathcal {A}}}}}_{C}\) contains that of \({\varvec{{{\mathcal {A}}}}}_{D}\), and so this n-cell cannot satisfy the condition (1) in Definition 4.22. Hence, it satisfies the condition (2) in this definition and so \({{\mathcal {B}}}_{C,i}\) is contained in the union of the translated i-mixed cells of \(S(\rho _{C})\). From here, we deduce that \({{\mathcal {B}}}_{D,i}\) is contained in the union of the translated i-mixed cells of \(S(\rho _{D})\) and as mentioned, the same happens for \(i=k\). Together with Proposition 4.16, this implies thatConsider the vector \({\varvec{\omega }}\in {\mathbb {R}}^{{\varvec{{{\mathcal {A}}}}}}\) defined by \(\omega _{j,a}=\theta _{k+1,j}(a)\) for \(j=0,\dots , n\) and \(a\in {{\mathcal {A}}}_{j}\). Since the chain is incremental, we have that \(\omega _{j,a}=0\) for \(j\ge k+1\) and \(a\in {{\mathcal {A}}}_{j}\) and soApplying Theorem 3.12 and Theorem 4.10 for the subsets \({{\mathcal {B}}}_{D}\) and \({{\mathcal {B}}}^{\circ }_{D}\), we deduce thatboth products being over the n-cells \(D'\) of \(S(\Theta _{{\varvec{\omega }}})=S(\theta _{k+1})\) that are contained in D. The equality in (4.16) then follows from the inductive hypothesis.

The second statement follows from the first equality in (4.17) applied to the case \(k=0\) and \(D=\Delta \). \(\square \)

We can extend the Canny–Emiris construction to a larger class of matrices, following an idea of the first author in [7]. The study of its properties can be done similarly as for the original formulation, and so we only indicate the modifications.

Let \(\psi :\Lambda \rightarrow {\mathbb {R}}\) be a convex piecewise affine function on a polytope such that the mixed subdivision \(S( \rho \boxplus \psi )\) on \(\Delta +\Lambda \) is tight and its \((n-1)\)-th skeleton does not contain any lattice point. The case treated in this paper corresponds to the situation when \(\Lambda =\{\delta \}\) and \(\psi \) takes any value at this point.

The index set is defined as \( {{\mathcal {B}}}= ( \Delta +\Lambda ) \cap M \). For each \(b\in {{\mathcal {B}}}\), there is a unique n-cell C of \(S(\rho \boxplus \psi )\) containing it, and we denote by \(C_{i}\in S(\rho _{i})\), \(i=0,\dots , n\), and \(B \in S(\psi )\) its components. The tightness condition implies that there is always an i such that \(\dim (C_{i})=0\), in which case \(C_{i}\) consists of a single lattice point of \({{\mathcal {A}}}_{i}\). We then setfor the largest of these i’s and the unique lattice point in the corresponding component, respectively. For \(b\in {{\mathcal {B}}}\) we have that \(b-a(b)+{{\mathcal {A}}}_{i(b)}\subset {{\mathcal {B}}}\) and so we can define a Sylvester map \(\Phi _{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\psi } \) and the corresponding Canny–Emiris matrix \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\psi }\) and determinant \(H_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\psi }\) in the same way as it was done before.

For \(i\in \{0,\dots , n\}\), we say that an n-cell C of \(S(\rho \boxplus \psi )\) is i-mixed if \(\dim (C_{j})=1\) for all \(j\ne i\). The nonmixed index subset \({{\mathcal {B}}}^{\circ } \) is the subset of \({{\mathcal {B}}}\) of lattice points lying in the n-cells that are not i-mixed for any i, and we denote by \({{\mathcal {E}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\psi }\) and \(E_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }},\psi }\) the corresponding principal submatrix and minor of the Canny–Emiris matrix.

Theorem 4.27 can then be extended to the statement thatwhenever the mixed subdivision \(S(\rho )\) is admissible in the sense of Definition 4.22, together with a factorization for \( E_{{\varvec{{{\mathcal {A}}}}}, {\varvec{\rho }},\psi }\) in terms of the n-cells of the second mixed subdivision in an admissible chain for \(S(\rho )\).

The extraneous factor \( E_{{\varvec{{{\mathcal {A}}}}}, {\varvec{\rho }}}\) does not depend on the choice of the admissible chain \(S(\theta _{0})\preceq \dots \preceq S(\theta _{n})\) but its factorization in Theorem 4.27 in principle depends on the second mixed subdivision \(S(\theta _{1})\). One can go further and refine this factorization using the subsequent mixed subdivisions in this chain. It would be interesting to exhibit concrete cases where different admissible chains produce different factorizations for this extraneous factor.

In addition, our factorization of \( E_{{\varvec{{{\mathcal {A}}}}}, {\varvec{\rho }}}\) consists of a product of extraneous factors of smaller systems in the same dimension, whereas the factorizations presented by Macaulay in [27, page 14] and by the first author in [7, (47)] consist of Canny–Emiris determinants and extraneous factors of systems in lower dimensions. It would be interesting to compare these approaches and put them into a more general framework.

Let \(n=2\), \({\varvec{d}}=(1,2,2)\) and \( m =|{\varvec{d}}|-n=3\). The corresponding general homogeneous polynomials areand the index set splits as \({{\mathcal {I}}}={{\mathcal {I}}}_{0}\sqcup {{\mathcal {I}}}_{1}\sqcup {{\mathcal {I}}}_{2}\) withThe matrix \({{\mathcal {M}}}_{{\varvec{d}}, m }\) is constructed by declaring that the row corresponding to each lattice point \({\varvec{c}}\in {{\mathcal {I}}}_{i}\), \(i=0,1,2\), consists of the coefficients of the polynomials \({\varvec{t}}^{{\varvec{c}}-d_{i}\, {\widehat{e}}_{i}}P_{i}\) in the monomial basis of the homogeneous part \({\mathbb {K}}[{\varvec{t}}]_{3}\). Hence, this matrix is written asWe have that \({{\mathcal {I}}}{\setminus } {{\mathcal {I}}}^{\mathrm{red}}= \{(1,2,0),(1,0,2)\}\) and soBy the identity in (5.3), \( {\text {Res}}_{{\varvec{d}}}\) is the quotient of the determinants of these matrices. It is an irreducible trihomogeneous polynomial in \({\mathbb {Z}}[{\varvec{\alpha }}, {\varvec{\beta }}, {\varvec{\gamma }}]\) of tridegree (4, 2, 2) having 234 monomial terms.

We have that \(\varphi =\sum _{i=0}^{n}\max \{0,\ell _{i}\}\) and the n-cells of \(S(\varphi )\) are the polytopes \(C_{I}\) for \(I\subset \{0,\dots , n\}\) with \(I,I^{\mathrm{c}}\ne \emptyset \). Moreover, for each I we have that

For \(n=2\), the mixed subdivision \(S(\varphi )\) has 6 maximal cells that, with notation as in Fig. 4, decompose as

The n-cells of \(S({\widehat{\tau }})\) are the polytopes \({\widehat{C}}_{I}\) for \(I\subset \{0,\dots , n\}\) with \(I,I^{\mathrm{c}}\ne \emptyset \). For each I, the vertices of \({\widehat{C}}_{I}\) are the lattice points \( {\widehat{v}}_{J,l}\) for \(J\subset I^{\mathrm{c}}\) and \(l\in I\).

For each i, the subdivision \(S({\widehat{\eta }}_{i})\) has two n-cells, one defined by \(z_{i}\ge d_{i}\) and the other by \(z_{i}\le d_{i}\). The n-cells of \(S( {\widehat{\tau }})\) are intersections of n-cells of these subdivisions and so they are of the form \({\widehat{C}}_{I}\) for \(I\subset \{0,\dots , n\}\).

We have that \(\sum _{i=0}^{n} z_{i}-d_{i}=0\) on \({\widehat{\Delta }}\), and so if either \(I= \emptyset \) or \(I^{\mathrm{c}}=\emptyset \) then the polytope \({\widehat{C}}_{I}\) reduces to the lattice point \({\varvec{d}}\). Otherwise, take \(0<\varepsilon < 1\) and consider the point \({\varvec{z}}=(z_{0},\dots , z_{n})\) defined by \(z_{j}= d_{j}+\frac{\varepsilon }{\#I}\) if \( j\in I\) and as \( z_{j}=d_{j}-\frac{\varepsilon }{\#I^{\mathrm{c}}}\) if \( j\in I^{\mathrm{c}}\). We have that \(|{\varvec{z}}|=|{\varvec{d}}|\) and that \(z_{j}>0\) for all j, and so \({\varvec{z}}\in {\text {ri}}({\widehat{\Delta }})\). Also for each i we have that \({\varvec{z}}\) lies in the relative interior of the corresponding n-cell of \(S({\widehat{\eta }}_{i})\). Hence, \({\widehat{C}}_{I}\) is n-dimensional in this case, proving the first claim.

Now let \(I\subset \{0,\dots , n\}\) with \(I,I^{\mathrm{c}}\ne \emptyset \). The vertices of \({\widehat{C}}_{I}\) are the intersections in this polytope of n of its supporting hyperplanes. These supporting hyperplanes are the zero set of one of the affine functionsTo compute the vertices, take disjoint subsets \(J\subset I^{\mathrm{c}}\) and \(K\subset \{0,\dots , n\}\) with \(\#J+\#K =n\). The intersection of the corresponding hyperplanes in the affine span of \({\widehat{\Delta }}\) isand it consists of the lattice pointfor the unique index l in the complement of \(J \cup K\).

When \(J\ne \emptyset \) we have that \({\widehat{v}}_{J,l} \in {\widehat{C}}_{I}\) if and only if \(l\in I\). When \(J=\emptyset \) we have that \({\widehat{v}}_{J,l}={\varvec{d}}\), which is also realized by taking \(l\in I\) and \(K=\{0, \dots , n\}{\setminus } \{l\}\), proving the second claim. \(\square \)

Let \(I\subset \{0,\dots , n\}\) with \(I,I^{\mathrm{c}}\ne \emptyset \) and set \(w_{I}=-\sum _{k\in I}{\widehat{e}}_{k}\). Then,

For \({\varvec{z}}\in {\widehat{\Delta }}\) we have that \({\widehat{\tau }}({\varvec{z}})=\sum _{j=0}^{n}\max \{0,z_{j}-d_{j}\}\), which readily implies thatwith equality if and only if \({\varvec{z}}\in {\widehat{C}}_{I}\). By the characterization of the cell \(\Gamma ({\widehat{\tau }}, w_{I})\) that follows from (2.4), this proves the statement in (1).

For each i, the vertices of \({\widehat{\Delta }}_{i}\) are the lattice points \(d_{i}\, {\widehat{e}}_{j}\), \(j=0,\dots , n\). For each j, we have that \( \langle w_{I}, d_{i}\, {\widehat{e}}_{j}\rangle = -d_{i}\) if \(j\in I\) and \(\langle w_{I}, d_{i}\, {\widehat{e}}_{j}\rangle = 0\) if \(j\in I^{\mathrm{c}}\), whereas \( {\widehat{\varphi }}_{i}(d_{i}\, {\widehat{e}}_{j})=0\) if \(j=i\) and \( {\widehat{\varphi }}_{i}(d_{i}\, {\widehat{e}}_{j})=d_{i}\) if \(j\ne i\). Hence,Then, the definition in (5.9) and the characterization in (2.4) easily imply that \({\widehat{C}}_{I,i}=\Gamma ({\widehat{\varphi }}_{i}, w_{I})\) which proves the first part of the statement in (2). Now let \({\varvec{z}}\in {\widehat{C}}_{I,i}\).  For \(i\!\in \! I\) the polytope \({\widehat{C}}_{I,i}\) consists of the single lattice point \(d_{i}\, {\widehat{e}}_{i}\) and so \({\widehat{\varphi }}_{i}({\varvec{z}})={\widehat{\varphi }}_{i}(d_{i}\, {\widehat{e}}_{i})= 0\). For \(i\in I^{\mathrm{c}}\) the vertices of \({\widehat{C}}_{I,i}\) are the lattice points \(d_{i}\, {\widehat{e}}_{j}\), \(j\in \{i\}\cup I\), and so \({\varvec{z}}=\sum _{j\in \{i\}\cup I} z_{j}\, e_{j}\). Hence, (5.10) gives that \({\widehat{\varphi }}_{i}({\varvec{z}})= -\langle w_{I}, {\varvec{z}}\rangle =\sum _{j\in I} z_{j}\), completing the proof of (2).

Let \({\widehat{v}}_{J,l}\) be the vertex of \({\widehat{C}}_{I}\) associated with a subset \(J\subset I^{\mathrm{c}}\) and an index \(l \in I\) as in (5.8). We have that \(d_{i}\, {\widehat{e}}_{l}\in {\widehat{C}}_{I,i}\) for all \(i\in I^{\mathrm{c}}\) and \(d_{i}\, {\widehat{e}}_{i}\in {\widehat{C}}_{I,i}\) for all i, and so \({\widehat{v}}_{J,l} \in \sum _{i=0}^{n}{\widehat{C}}_{I,i}\). Since this holds for all J and l, Lemma 5.4 implies thatFrom (2) and Proposition 2.1(1), we deduce that the \({\widehat{C}}_{I,i}\)’s are the components of the cell \(\Gamma ({\widehat{\varphi }}, w_{I})\) of \(S({\widehat{\varphi }})\) and by Proposition 2.1(2), we have that \(\sum _{i=0}^{n}{\widehat{C}}_{I,i}\) is a cell of this mixed subdivision. On the other hand, the \({\widehat{C}}_{I}\)’s are polytopes that cover \({\widehat{\Delta }}\) and so the inclusion in (5.11) is an equality, as stated in (3).

Now let \({\varvec{z}}=(z_{0},\dots , z_{n})\in {\widehat{C}}_{I}\) and \({\varvec{z}}_{i}\in {\widehat{C}}_{I,i}\), \(i=0,\dots ,n\), such that \({\varvec{z}}=\sum _{i=0}^{n}{\varvec{z}}_{i}\). Necessarily \({\varvec{z}}_{i}=d_{i}\, {\widehat{e}}_{i}\) for \(i\in I\) and sowhere the first equality follows from (1) and the two last from (2) and Proposition 2.1(3), respectively. Hence, \({\widehat{\tau }}\) and \({\widehat{\varphi }}\) coincide on \({\widehat{C}}_{I}\) and so on the whole of \({\widehat{\Delta }}\), proving (4). \(\square \)

Consider the projection \(\pi :{\mathbb {R}}^{n+1}\rightarrow {\mathbb {R}}^{n}\) defined byThis linear map induces isomorphisms between \({\widehat{\Delta }}\) and \(\Delta \) and between \({\widehat{\Delta }}_{i}\) and \(\Delta _{i}\) for each i, and it also satisfies that \( {\widehat{\varphi }}_{i}=\varphi _{i}\circ \pi \) for each i.

For \({\varvec{z}}\in {\widehat{\Delta }}\) and \({\varvec{x}}=\pi ({\varvec{z}})\in \Delta \), the condition that \( {\varvec{z}}=\sum _{i=0}^{n}{\varvec{z}}_{i} \) with \({\varvec{z}}_{i}\in {\widehat{\Delta }}_{i}\) translates into \({\varvec{x}}=\sum _{i=0}^{n} {\varvec{x}}_{i}\) with \({\varvec{x}}_{i}\in \Delta _{i}\), and vice versa. Precisely, if the first condition holds then so does the second with \({\varvec{x}}_{i}=\pi ({\varvec{z}}_{i})\) and conversely, if the second condition holds then so does the first with \({\varvec{z}}_{i}\) defined as the only preimage of \({\varvec{x}}_{i}\) in \({\widehat{\Delta }}_{i}\).

We deduce from Lemma 5.5(4) that \(\varphi \circ \pi = {\widehat{\varphi }}={\widehat{\tau }}\), and the statement then follows from Lemmas 5.4 and 5.5. \(\square \)

Proposition 5.2(3) implies that the mixed subdivision \(S(\varphi )\) is tight only when \(n\le 2\).

We have that \(\theta _{k} =\sum _{i=0}^{k-1} \max \{0,\ell _{i}\} + \sum _{i=k}^{n}\ell _{i} \) and the n-cells of the mixed subdivision \(S(\theta _{k})\) are the polytopes \(C_{k,I}\) for \(I\subset \{0,\dots , k-1\}\). Moreover, for each I

Denote with a hat the corresponding objects in \({\mathbb {R}}^{n+1}\) as it was previously done for the study of \(\varphi \), and consider also the convex piecewise affine function on \(\Delta \) defined asThe proof of these properties is a direct generalization of that for Proposition 5.2 and so we only indicate the main steps:Finally, the obtained results are brought back to \({\mathbb {R}}^{n}\) via the projection \(\pi \), as in the proof of Proposition 5.2. \(\square \)

We have that \(S(\theta _{0})\preceq \cdots \preceq S(\theta _{n})\) is an incremental chain of mixed subdivisions of \(\Delta \) with \(S(\theta _{n}) \preceq S(\varphi )\) that satisfies the conditions in Proposition 4.23. In particular, this incremental chain is admissible for any tight mixed subdivision of \(\Delta \) that refines \(S(\varphi )\).

For each \(I\subset \{0,\dots , n\} \) with \(I,I^{\mathrm{c}}\ne \emptyset \) we have thatPropositions 5.2 and 5.7 then imply that \(S(\varphi )\succeq S(\theta _{n})\). Similarly, for \(k\in \{1,\dots , n\}\) and each \(I\subset \{0,\dots , k-1\}\) we have thatand Proposition 5.7 implies that \(S(\theta _{k})\!\succeq \! S(\theta _{k-1})\). Hence, \( S(\theta _{0})\preceq \cdots \preceq S(\theta _{n}) \!\preceq \! S(\varphi )\). Since \(\theta _{k,i}=0|_{\Delta _{i}}\) for all k and \(i\ge k\), the chainis incremental.

For \(k=0,\dots , n\), let \(I\subset \{0,\dots , k-1\}\). If \(I\ne \emptyset \) then for the n-cell \(C_{I}\) of \(S(\theta _{k})\), each component \(C_{I,i}\) with \(i\in I\) consists of a lattice point and so this component verifies the condition (2) in Proposition 4.23. Else \(I=\emptyset \) and so \(C_{k,I,i}=d_{i} {\text {conv}}( e_{i}, \{e_{j}\}_{j\in \{k,\dots , n\}})\) if \( i<k\) and \( C_{k,I,i}= d_{i} {\text {conv}}( \{e_{j}\}_{j\in \{k,\dots , n\}})\) if \( i\ge k\). For \(i=0,\dots , k-1\), consider the nonzero vector \(w_{i}\in {\mathbb {R}}^{n}\) defined as \(w_{i}=\sum _{j=1}^{n}e_{j}\) if \(i=0\) and as \(w_{i}=e_{i}\) if \(i>0\). For all \(j\ne i\), we have that \(C_{k,I,j}\) lies in a hyperplane that is parallel to \(w_{i}^{\bot }\) and soHence, \(C_{k,I}\) satisfies the condition (3) in Proposition 4.23. The last statement is a direct consequence of that proposition. \(\square \)

The morphism of algebras \( \pi _{*} :{\mathbb {K}}[{\varvec{t}}]\rightarrow {\mathbb {K}}[{\varvec{s}}]\) defined by \(\pi _{*}(t_{0})= 1\) and \(\pi _{*}(t_{i})= s_{i}\), \(i=1,\dots , n\), induces a commutative diagramand bijections between the monomial bases of \( \bigoplus _{i=0}^{n}T_{i}\) and \( \bigoplus _{i=0}^{n}V_{i}\), and between those of T and V. In particular, \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}={{\mathcal {M}}}_{{\varvec{d}}}\) and \({{\mathcal {E}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}={{\mathcal {N}}}_{{\varvec{d}}}\).

Let C be an n-cell of \(S(\rho )\). Let \(I\subset \{0,\dots , n\}\) with \(I, I^{\mathrm{c}}\ne \emptyset \) such that \(C\subset C_{I}\) and \(i\in \{0, \dots , n\}\). Then,

Since \(S(\rho )\succeq S(\varphi )\), we have that \(C_{i}\subset C_{I,i}\). If \(i\in I\) then (5.7) implies that \(C_{i}=\{d_{i}\, e_{i}\}\). Conversely, for \(i \in I^{\mathrm{c}}\) consider the vector \(w_{i}\in {\mathbb {R}}^{n}\) defined as \(w_{i}=\sum _{j=1}^{n}e_{j}\) if \(i=0\) and as \(w_{i}=e_{i}\) if \(i>0\). For all \(j \ne i\) we have that \(C_{I,j}\) lies in a hyperplane that is parallel to \(w_{i}^{\bot }\), and so does \(C_{j}\). Hence,proving (1). The statement in (2) is a direct consequence of that in (1): C is i-mixed if and only if \(C_{I,i}\) is the unique component of \(C_{I}\) of dimension 0, which is equivalent to the fact that \(I=\{i\}\). \(\square \)

\( {{\mathcal {B}}}=\{{\varvec{b}}\in {\mathbb {N}}^{n} \mid |{\varvec{b}}| \le |{\varvec{d}}|-n\}\).

Let \({\varvec{b}}=(b_{1},\dots , b_{n}) \in {\mathbb {Z}}^{n}\). Then, \({\varvec{b}}\in \Delta +{\varvec{\delta }}\) if and only if \(b_{i}-\delta _{i}\ge 0\) for all i and \(|{\varvec{b}}- {\varvec{\delta }}|\le |{\varvec{d}}|\). Since \({\varvec{b}}\) is a lattice point, \(\delta _{i}+1 > 0\) for all i and \(\sum _{i=1}^{n}(\delta _{i}+1) <1\), these conditions are equivalent to \(b_{i}\ge 0\) for all i and \(|{\varvec{b}}|\le |{\varvec{d}}|-n\), proving the lemma. \(\square \)

Let \({\varvec{b}}=(b_{1},\dots , b_{n})\in {{\mathcal {B}}}\) and set \(b_{0}=|{\varvec{d}}|-n-|{\varvec{b}}|\). Then,

With notation as in (5.6), we have that \({\varvec{b}}\in C_{I}+{\varvec{\delta }}\) if and only if \(\ell _{i}({\varvec{b}}-{\varvec{\delta }}) \ge 0\) for \(i\in I\) and \(\ell _{i}({\varvec{b}}-{\varvec{\delta }}) \le 0\) for \(i\in I^{\mathrm{c}}\). Arguing as in the proof of Lemma 5.11, we deduce that these conditions are equivalent to \(b_{i}\ge d_{i}\) for all \(i\in I\) and \(b_{i}< d_{i}\) for all \(i\in I^{\mathrm{c}}\), as stated in (1).

The statements in (2) and (3) follow from that in (1) together with Lemma 5.10. \(\square \)

Let \(\pi :{\mathbb {Z}}^{n+1}\rightarrow {\mathbb {Z}}^n\) be the linear map given by \(\pi (c_{0},\dots , c_{n})= (c_{1},\dots , c_{n})\), so that \(\pi _{*}({\varvec{t}}^{{\varvec{c}}})={\varvec{s}}^{\pi ({\varvec{c}})}=s_{1}^{c_{1}}\dots s_{n}^{c_{n}}\) for all \( {\varvec{c}}=(c_{0},\dots , c_{n})\in {\mathbb {N}}^{n+1}\).

By Lemma 5.11, \(\pi \) induces a bijection between the index sets \({{\mathcal {I}}}\) and \({{\mathcal {B}}}\), and so the morphism of algebras \(\pi _{*}\) gives a bijection between the monomial bases of T and of V. Similarly, by Lemma 5.12(2) \(\pi \) also induces a bijection between \({{\mathcal {I}}}_{i}\) and \({{\mathcal {B}}}_{i}\) for each i, and so \(\pi _{*}\) gives a bijection between the monomial bases of \( \bigoplus _{i=0}^{n}T_{i}\) and \( \bigoplus _{i=0}^{n}V_{i}\), proving the second claim. Moreover, for \({\varvec{c}}\in {{\mathcal {I}}}\) we have thatwhich shows the commutativity of the diagram. The last claim is a direct consequence of the two previous. \(\square \)

\({\text {Res}}_{{\varvec{d}}}= {\det ({{\mathcal {M}}}_{{\varvec{d}}})}/{\det ({{\mathcal {N}}}_{{\varvec{d}}})}\).

By Proposition 5.8, the mixed subdivision \(S(\rho )\) is admissible. Theorem 1.3 and Proposition 5.9 then imply thatThe sign can be determined by considering the evaluation of both sides of this equality at the coefficients of systems of polynomials \(t_{i}^{d_{i}}\), \(i=0,\dots , n\).

Consider again the case when \(n=2\), \({\varvec{d}}=(1,2,2)\) and \(m=3\). Then,By Proposition 5.2(3), the mixed subdivision \(S(\varphi )\) is tight and so we can take \(\rho _{i}=\varphi _{i}\), \(i=0, 1,2\). We also choose \({\varvec{\delta }}=(-\frac{2}{3}, -\frac{3}{4}) \in {\mathbb {R}}^{2}\).

As shown in Fig. 6, the index set \({{\mathcal {B}}}\) splits as \({{\mathcal {B}}}={{\mathcal {B}}}_{0}\sqcup {{\mathcal {B}}}_{1}\sqcup {{\mathcal {B}}}_{2}\) withand the row content function assigns to the elements of \({{\mathcal {B}}}_{i}\), \(i=0, 1, 2\), the vertices \((0,0)\in \Delta _{0}\), \((2,0)\in \Delta _{1}\), \((0,2)\in \Delta _{2}\), respectively. Moreover, the elements of \({{\mathcal {B}}}\) lying in the translated nonmixed 2-cells are (2, 0) and (0, 2).

The Canny–Emiris matrix \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) and its principal submatrix \({{\mathcal {E}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\), respectively, coincide with the Macaulay matrices \({{\mathcal {M}}}_{{\varvec{d}}}\) and \({{\mathcal {N}}}_{{\varvec{d}}}\) in Example 5.1, in agreement with Proposition 5.9.

The Macaulay matrix \({{\mathcal {M}}}_{{\varvec{d}}}\) cannot be produced by a mixed subdivision admitting a tight incremental chain (Definition 2.4). This can be shown by inspecting the different mixed subdivisions of \(\Delta \) that allow such an incremental chain and verifying that none of them coincides with \(S(\rho )\). One of these incremental chains for the case when \(n=2\) and \({\varvec{d}}=(1,3,2)\) is shown in Example 2.7.

Let notation be as in Example 5.14 and instead of the \(\varphi _{i}\)’s, consider the affine functions \(\rho _{i}:\Delta _{i}\rightarrow {\mathbb {R}}\), \(i=0,1,2\), defined byLet \(\rho :\Delta \rightarrow {\mathbb {R}}\) be their inf-convolution. The mixed subdivision \(S(\rho )\) of \(\Delta \) is tight and has 6 maximal cells as shown in Fig. 7, that decompose as

The index set and row function corresponding to the data \({\varvec{{{\mathcal {A}}}}}=({{\mathcal {A}}}_{0}, {{\mathcal {A}}}_{1}, {{\mathcal {A}}}_{2})\), \({\varvec{\rho }}=(\rho _{0},\rho _{1},\rho _{2})\) and \({\varvec{\delta }}\) are equal to those in Example 5.14, and so the Canny–Emiris matrix \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) also coincides with that in (5.4). However, the translated nonmixed cells of \(S(\rho )\) differ from those in Example 5.14. Their lattice points are (3, 0) and (0, 3) and the corresponding principal submatrix isThe determinant of this matrix does not divide that of \({{\mathcal {H}}}_{{\varvec{{{\mathcal {A}}}}},{\varvec{\rho }}}\) and so the formula in Theorem 1.3 does not hold in this case. In particular, \(S(\rho )\) is not admissible.

Indeed, this latter observation can be verified directly: letbe an incremental chain of mixed subdivisions of \(\Delta \) with \(S(\theta _{2})\preceq S(\rho )\) and for each \(k=0,1,2\) let \(\theta _{k,i}:\Delta _{i}\rightarrow {\mathbb {R}}\), \(i=0,1,2\), be the corresponding family of convex piecewise affine functions.

If \(\theta _{1,0}:\Delta _{0}\rightarrow {\mathbb {R}}\) is not constant, then \(S(\theta _{1})\) has a cell that is a translate of the triangle \( \Delta _{1}+\Delta _{2}= \{(x_{1},x_{2}) \in ({\mathbb {R}}_{\ge 0})^{2} \mid x_{1}+x_{2} \le 4\}\) which is not compatible with the assumption that \(S(\rho )\) is a refinement of \(S(\theta _{1})\), as it can be verified in Fig. 6. We deduce that \(\theta _{1,0}:\Delta _{0}\rightarrow {\mathbb {R}}\) is constant, but in this case \(S(\theta _{1})\) is the trivial mixed subdivision of \(\Delta \) and this incremental chain does not verify the conditions in Definition 4.22 for \(k=1\), and so it is not admissible.