Ideal Membership in Polynomial Rings over the Integers

Abstract

We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials $f_0,f_1,...,f_n\in\Z[X]$, where $X=(X_1,...,X_N)$ is an $N$-tuple of indeterminates, are there $g_1,...,g_n\in\Z[X]$ such that $f_0=g_1f_1+...+g_nf_n$? We show that the degree of the polynomials $g_1,...,g_n$ can be bounded by $(2d)^{2^{O(N^2)}}(h+1)$ where $d$ is the maximum total degree and $h$ the maximum height of the coefficients of $f_0,...,f_n$. Some related questions, primarily concerning linear equations in $R[X]$, where $R$ is the ring of integers of a number field, are also treated.