A dynamical solution of Kronecker's problem

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".