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 Diskr. Mat., 2004, Volume 16, Issue 1, Pages 52–78 (Mi dm142)

The standard basis of a polynomial ideal over a commutative Artinian chain ring

E. V. Gorbatov

Abstract: We construct a standard basis of an ideal of the polynomial ring $R[X]=R[x_1,\ldots,x_k]$ over commutative Artinian chain ring $R$, which generalises a Gröbner base of a polynomial ideal over fields. We adopt the notion of the leading term of a polynomial suggested by D. A. Mikhailov and A. A. Nechaev, but using the simplification schemes introduced by V. N. Latyshev. We prove that any canonical generating system constructed by D. A. Mikhailov and A. A. Nechaev is a standard basis of the special form. We give an algorithm (based on the notion of $S$-polynomial) which constructs standard bases and canonical generating systems of an ideal. We define minimal and reduced standard bases and give their characterisations. We prove that a Gröbner base $\chi$ of a polynomial ideal over the field $\bar R=R/\operatorname{rad}(R)$ can be lifted to a standard basis of the same cardinality over $R$ with respect to the natural epimorphism $\nu\colon R[X]\to \bar R[X]$ if and only if there is an ideal $I\triangleleft R[X]$ such that $I$ is a free $R$-module and $\bar{I}=(\chi)$.
The research was supported by the Russian Foundation for Basic Research, grant 02-01-00218, and by the President of the Russian Federation program of support of leading scientific schools, grant 1910.2003.1.

DOI: https://doi.org/10.4213/dm142

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English version:
Discrete Mathematics and Applications, 2004, 14:1, 75–101

Bibliographic databases:

UDC: 512.8

Citation: E. V. Gorbatov, “The standard basis of a polynomial ideal over a commutative Artinian chain ring”, Diskr. Mat., 16:1 (2004), 52–78; Discrete Math. Appl., 14:1 (2004), 75–101

Citation in format AMSBIB
\Bibitem{Gor04} \by E.~V.~Gorbatov \paper The standard basis of a polynomial ideal over a commutative Artinian chain ring \jour Diskr. Mat. \yr 2004 \vol 16 \issue 1 \pages 52--78 \mathnet{http://mi.mathnet.ru/dm142} \crossref{https://doi.org/10.4213/dm142} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2069989} \zmath{https://zbmath.org/?q=an:1096.13034} \transl \jour Discrete Math. Appl. \yr 2004 \vol 14 \issue 1 \pages 75--101 \crossref{https://doi.org/10.1515/156939204774148820} 

• http://mi.mathnet.ru/eng/dm142
• https://doi.org/10.4213/dm142
• http://mi.mathnet.ru/eng/dm/v16/i1/p52

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. E. V. Gorbatov, “Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences”, J. Math. Sci., 139:4 (2006), 6672–6707
2. E. V. Gorbatov, “Multiplicative orders on terms”, J. Math. Sci., 152:4 (2008), 517–521
3. Kuijper M., Schindelar K., “The predictable leading monomial property for polynomial vectors over a ring”, 2010 IEEE International Symposium on Information Theory, IEEE International Symposium on Information Theory, 2010, 1133–1137
4. Kuijper M., Schindelar K., “Minimal Grobner bases and the predictable leading monomial property”, Linear Algebra and Its Applications, 434:1 (2011), 104–116
5. Kuijper M., Pinto R., “An iterative algorithm for parametrization of shortest length linear shift registers over finite chain rings”, Des. Codes Cryptogr., 83:2 (2017), 283–305
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