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A dynamical solution of Kronecker's problem

Authors: Ihsen Yengui

Published in: Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006)

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".

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Ihsen Yengui. A dynamical solution of Kronecker's problem. In Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, Volume 5021, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.3},
  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}
}

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A dynamical solution of Kronecker's problem

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