A dynamical solution of Kronecker's problem

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 Z[X1, . . . , Xn]. 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 Gröbner bases” of polynomial ideals over a principal domain. As application, I give an alternative dynamical solution to “Kronecker’s problem”.