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Applicable Algebra in Engineering, Communication and Computing
Published in: Applicable Algebra in Engineering, Communication and Computing 6/2022

07-11-2022 | Original Paper

De Nugis Groebnerialium 6: Rump, Ufnarovski, Zacharias

Authors: Michela Ceria, Ferdinando Mora

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 6/2022

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Abstract

Moved by a question posed us by Wolfgang Rump, we investigate the Rump ideal \({\mathbb {I}}(p^2-pq+qp)\subset {\mathbb {Z}}\langle q,q^{-1}, p\rangle \) and we show, this way, the power of Zacharias representation.
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Appendix
Available only for authorised users
Footnotes
1
and it is computable via Buchberger reduction.
 
2
\(p\in \mathsf{N}_{00}\) for \(n=1\) and, for each \(i\in {\mathbb {Z}}\setminus \{0\}, q^i\in \mathsf{N}_{ii}\) for \(n=0\)
 
3
The value \(\xi _B\) is indeed the coefficient of B making it reducible by an \(f_i \in G\).
 
4
By symmetry, \(\mathfrak {S}\left( S(a,b;pq^{a-1},1;1,q^{b-1}p)\right) =0\) also if we choose \(\mathbf{T}(h)= q^ap q^{b-1}p\).
 
5
Note that \(h_{15,21}=12pq^{35}p-5pq^{14}pq^{21}-\mathbf{7q^{15}pq^{20}p}\) realted to the parameters \(a=15, b=21,\alpha =7, \beta =5, \mathrm{lcm}(a,b)=105, \gcd (a,b)=3\)
 
6
At the conference CoCoA I held in Genoa in May 1986, where also [7] was presented.
 
7
A strategy for choosing the next S-polynomial in a Gebauer–Möller set is said fair when each pair is chosen within a finite number of choices and thus cannot remain unprocessed forever.
 
Literature
1.
2.
Buchberger, B.: A criterion for detecting unnecessary reduction in the construction of Gröbner bases. L. N. Comput. Sci. 72, 3–21 (1979)MATH
3.
Buchberger, B.: Introduction to Göbner Bases. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Application, pp. 3–31. Cambridge University Press, Cambridge (1998)CrossRefMATH
4.
Ceria, M., Mora, T.: Buchberger-Weispfenning theory for effective associative rings. J. Symb. Comput. special issue for ISSAC, 83, 112–146 (2015)
5.
Cioffi, F., Roggero, M.: Flat families by strongly stable ideals and a generalization of Gröbner bases. J. Symb. Comput. 49, 1070–1084 (2011)CrossRefMATH
6.
Cojocaru, S., Ufnarovski, V.: Noncommuatative Gröbner basis, Hilbert series, Anick’s resolution and BERGMAN under MS-DOS. Comput. Sci. J. Moldova 3, 24–39 (1995)
7.
Gebauer, R., Möller, H.M.: On an Installation of Buchbgerger’s Algorithm. J. Symb. Comput. 6, 275–286 (1988)CrossRefMATH
8.
Gebauer R., Möller H.M.: A fast Variant of Buchbgerger’s Algorithm, Preprint (1985)
9.
Green, E., Mora, T., Ufnarovski, V.: The non-commutative Gröbner freaks. In: Bronstein, M., Grabmeier, J., Weispfenning, V. (eds.) Symbolic Rewriting Techniques, pp. 93–104. Birkhäuser, Basel (1998)CrossRef
10.
Gröbner W.: Algebraische Geometrie I, Bibliographische Institut (1968)
11.
Janet, M.: Sur les systèmes d’équations aux dérivées partielles. J. Math. Pure et Appl. 3, 65–151 (1920)MATH
12.
Levandovskyy V.G.,Studzinski G., Schnitzler, B.: Enhanced computations of Gröbner bases in free algebras as a new application of the Letterplace Paradigm. In: Proc. ISSAC’13 (2013), pp. 259–266, A.C.M
13.
Månsson, J.: On the computation of Hilbert series and Poincare series for algebras with infinite Gröbner bases. Comput. Sci. J. Moldova 8(1), 42–63 (2000)MathSciNetMATH
14.
15.
Mårtensson, K.: An algorithm to detect regular behaviour of binomial Gröbner Basis rational language. Master’s Thesis, Lund University (2006)
16.
Map W.: De nugis curialium; Courtiers’ Trifles, ed. and trans. M. R. James. Revised by C. N.’ L. Brooke and R. A. B. Mynors. (Oxford Medieval Texts.) New York: Clarendon Press, Oxford University Press (1983)
17.
Möller, H.M.: On the construction of Gröbner bases using syzygies. J. Symb. Comput. 6, 345–359 (1988)CrossRefMATH
18.
Mora, T.: An introduction to commutative and non-commutative Groebner bases. Theor. Comput. Sci. 134, 131–173 (1994)CrossRefMATH
19.
Mora, F.: De Nugis Groebnerialium 1: Eagon, Northcott, Groebner. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Application, pp. 434–447. Cambridge University Press, Cambridge (1998)CrossRefMATH
20.
Mora, F.: De Nugis Groebnerialium 2: Applying Macaulay’s trick in order to easily write a Groebner basis. Appl. Algebra Eng. Commun. Comput. 13(6), 437–446 (2003)CrossRefMATH
21.
Mora, F.: De Nugis Groebnerialium 3: Solitaire, Reiß, Gröbner. Rendiconti Del Seminario Matematico 78(1), 43–56 (2021)MATH
22.
Mora F.: De Nugis Groebnerialium 4: Zacharias, Spears, Möller. In: Proc. ISSAC’15 (2015), pp. 191–198, ACM
23.
Mora, F.: De Nugis Groebnerialium 5: Noether, Macaulay, Jordan. In: Proceedings of the 24\(^{th}\) Conference on Applications of Computer Algebra (ACA 2018 Santiago de Compostela) Rendiconti Del Seminario Matematico, vol. 78(1), pp. 43–56 (2018)
24.
Mora, T.: Solving Polynomial Equation Systems. 4 Vols., Cambridge University Press, I (2003), II (2005), III (2015), IV (2016)
25.
26.
27.
28.
Ollivier, F.: Canonical bases: reletions with standard bases, finiteness conditions and application to tame automorphisms. Progress Math. 94, 379–400 (1990)MathSciNet
29.
Pritchard F.L.: A syzygies approach to non-commutative Gröbner bases, Preprint (1994)
30.
31.
Rump, W.: Degenerate involutive set-theoretic solutions to the Yang-Baxter equation. Prerint (2021)
32.
Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrass’schen Divisionsatz. Diplomarbait, Hamburg (1980)
33.
Schreyer, F.O.: A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math. 421, 83–123 (1991)MathSciNetMATH
34.
Zacharias G.: Generalized Gröbner bases in commutative polynomial rings, Bachelor’s thesis, M.I.T. (1978)
Metadata
Title
De Nugis Groebnerialium 6: Rump, Ufnarovski, Zacharias
Authors
Michela Ceria
Ferdinando Mora
Publication date
07-11-2022
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 6/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-022-00583-2

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