License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.05021.3
URN: urn:nbn:de:0030-drops-2889
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2006/288/
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Yengui, Ihsen

A dynamical solution of Kronecker's problem

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05021.YenguiIhsen.Paper.288.pdf (0.2 MB)

Abstract

In this paper,
I present a new decision procedure for the ideal
membership problem for polynomial rings over principal domains
using discrete valuation domains. As a particular case, I solve a
fundamental algorithmic question in the theory of multivariate
polynomials over the integers called ``Kronecker's problem", that
is the problem of finding a decision procedure for the ideal
membership problem for $mathbb{Z}[X_1,ldots, X_n]$. The
techniques utilized are easily generalizable to Dedekind domains.
In order to avoid the expensive complete factorization in the
basic principal ring, I introduce the notion of ``dynamical
Gr"obner bases" of polynomial ideals over a principal domain. As
application, I give an alternative dynamical solution to
``Kronecker's problem".

BibTeX - Entry

@InProceedings{yengui:DagSemProc.05021.3,
  author =	{Yengui, Ihsen},
  title =	{{A dynamical solution of Kronecker's problem}},
  booktitle =	{Mathematics, Algorithms, Proofs},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{5021},
  editor =	{Thierry Coquand and Henri Lombardi and Marie-Fran\c{c}oise Roy},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2006/288},
  URN =		{urn:nbn:de:0030-drops-2889},
  doi =		{10.4230/DagSemProc.05021.3},
  annote =	{Keywords: Dynamical Gr\~{A}ƒ\^{A}¶bner basis, ideal membership problem, principal domains}
}

Keywords: Dynamical Gröbner basis, ideal membership problem, principal domains
Collection: 05021 - Mathematics, Algorithms, Proofs
Issue Date: 2006
Date of publication: 16.01.2006

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