2003/05/12 by Matthias Aschenbrenner, Aschenbrenner, Matthias · 2 citations
Computer Science · Mathematics · #11C08 #13P10 #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AC #math.NT #msc:11C08 #msc:13P10
paper · pdf · doi:10.48550/arxiv.math/0305172
34 pages
openalex publication_date 2003/05/12 · arxiv created 2003/06/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0,f1,...,fn∈\Z[X], where X=(X1,...,XN) is an N-tuple of indeterminates, are there g1,...,gn∈\Z[X] such that f0=g1f1+...+gnfn? We show that the degree of the polynomials g1,...,gn can be bounded by (2d)^2O(N2)(h+1) where d is the maximum total degree and h the maximum height of the coefficients of f0,...,fn. Some related questions, primarily concerning linear equations in R[X], where R is the ring of integers of a number field, are also treated.