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A dynamical solution of Kronecker's problem

Author Ihsen Yengui



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Ihsen Yengui

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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)
https://doi.org/10.4230/DagSemProc.05021.3

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".
Keywords
  • Dynamical Gröbner basis
  • ideal membership problem
  • principal domains

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