To make the phrase “vary favourably” concrete, assume that
i ranges through the objects of a category
C and
\({\mathbb A}(i)\) is a finite-dimensional affine space varying functorially with
i. This means that for each
\(\pi \in {\text {Hom}}_C(i,j)\) we have a linear map
\({\mathbb A}(\pi ):{\mathbb A}(i) \rightarrow {\mathbb A}(j)\) such that
\({\mathbb A}({\text {id}}_i)={\text {id}}_{{\mathbb A}(i)}\) and
\({\mathbb A}(\sigma \circ \pi )={\mathbb A}(\sigma ) \circ {\mathbb A}(\pi )\) for
\(\sigma \in {\text {Hom}}_C(j,k)\). Suppose that
\({\mathbb A}'(i)\) is another affine space depending functorially on
\(i \in C\), and finally that, for any
\(i \in C\), we have a polynomial map
\(\varphi _i:{\mathbb A}(i) \rightarrow {\mathbb A}'(i)\) such that the following diagram commutes for every
\(\pi \in {\text {Hom}}_C(i,j)\):
In this case, if
f is a polynomial equation for the image closure
\(\overline{{\text {im}}(\varphi _j)}\) of
\(\varphi _j\), then
\(f \circ {\mathbb A}'(\pi )\) is a polynomial equation for
\(\overline{{\text {im}}(\varphi _i)}\)—and, since we assumed that
\({\mathbb A}'(\pi )\) is linear, of the same degree. We then ask:
(1)
Do there exist finitely many \(j \in C\) such that the equations for those \(\overline{{\text {im}}(\varphi _j)}\), by pulling back along the linear maps \({\mathbb {A}}'(\pi )\) for all relevant \(\pi \), define \(\overline{{\text {im}}(\varphi _i)}\) for all \(i \in C\)?
(2)
If so, does there exist an algorithm for computing these finitely many j?
A well-known case where the answer to the two questions above is “yes” is that where
C is the opposite category
\(\mathbf {FI^{op}}\) of the category
\(\textbf{FI}\) of finite sets with injections,
\({\mathbb A}(i)=({\mathbb A}^n)^i\),
\({\mathbb A}'(i)=({\mathbb A}^{n'})^i\) for some fixed
n and
\(n'\), and the maps
\({\mathbb A}(i) \rightarrow {\mathbb A}(j), {\mathbb A}'(i) \rightarrow {\mathbb A}'(j)\) corresponding to an injection
\(j \rightarrow i\) are the canonical projections. In that case, it is known that the kernel of the
\(\textbf{FI}\)-homomorphism
\(\varphi ^*\) dual to the
\(\mathbf {FI^{op}}\)-polynomial map
\(\varphi \) is finitely generated and can be computed using a version of Buchberger’s algorithm, and this has been applied to problems in algebraic statistics [
4,
5,
15‐
17]. The setting discussed in the current paper is of a very different flavour in that it involves continuous symmetries rather than discrete symmetries, and it is also significantly more complicated. One cause for trouble is that in the setting below, we do not actually know whether the kernel of the relevant algebra homomorphism is finitely generated, and so we have to settle for finding set-theoretic equations for the
\(\overline{{\text {im}}(\varphi _i)}\).