An algorithm to compute the set of characteristics of a system of polynomial equations over the integers.
The question considered in the paper under review is: For a given system of multivariate polynomials with integer coefficients, what are the possible characteristics of fields where this system is solvable? The problem is prompted by and has applications in matroid theory. The authors use the Gröbner bases theory over the integers to provide an algorithm to find this set \(\chi (F)\) for any specific system \(F\). Moreover, they demonstrate that either \(\chi (F)\) is finite and does not contain \(0\), or \(0 \in \chi (F)\) and only finitely many primes are missing from \(\chi (F)\). In the final appendix of the paper it is shown that the result can be proved in a more abstract context (constructible sets of prime ideals in affine algebras over Noetherian domains of Krull dimension 1).
solvable system of multivariate polynomials
scientific article; zbMATH DE number 1965168