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Ideal Membership in Polynomial Rings over the Integers
 J. Amer. Math. Soc
"... Abstract. 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 Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of th ..."
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Abstract. 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 Ntuple 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.
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1