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A dynamical solution of Kronecker's problem Yengui, Ihsen Dynamical Gröbner basis ideal membership problem principal domains 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". Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ihsen Yengui 2006 Is Part Of Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.05021.3 urn:nbn:de:0030-drops-2889 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.3 eng https://creativecommons.org/licenses/by/4.0/legalcode