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Yengui, Ihsen

A dynamical solution of Kronecker's problem

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


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

  author =	{Ihsen Yengui},
  title =	{A dynamical solution of Kronecker's problem},
  booktitle =	{Mathematics, Algorithms, Proofs},
  year =	{2006},
  editor =	{Thierry Coquand and Henri Lombardi and Marie-Fran{\c{c}}oise Roy},
  number =	{05021},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{},
  annote =	{Keywords: Dynamical Gr{\"o}bner basis, ideal membership problem, principal domains}

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

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