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 $f_0,f_1,...,f_n\in \mathbb{Z}[X]$, where $X=(X_1,...,X_N)$ is an $N$-tuple of indeterminates, are there $g_1,...,g_n\in \mathbb{Z}[X]$ such that $f_0=g_1{f}_1+...+g_n{f}_n$? 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.

34 pages

Cited by in corpus (1)

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