Using Semicontinuity for Standard Bases Computations

TL;DR

This paper introduces a new algorithm leveraging semicontinuity for faster standard basis computations of 0-dimensional ideals in power series and localized polynomial rings, especially effective over rings of integers or parameterized polynomial rings.

Contribution

The authors develop an algorithm that speeds up standard basis calculations by using semicontinuity and specialization techniques, applicable over various coefficient rings.

Findings

Algorithm significantly reduces computation time with coefficient swell.

Correctness is guaranteed by a semicontinuity theorem.

Implemented in Singular, it demonstrates practical efficiency.

Abstract

We present new results and an algorithm for standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring in finitely many variables over a field K. The algorithm provides a significant speed up if K is the quotient field of a Noetherian integral domain A, when coefficient swell occurs. The most important special cases are perhaps when A is the ring of integers resp. when A is a polynomial ring over some field in finitely many parameters. Given I as an ideal in the polynomial ring over A, we compute first a standard basis modulo a prime number p, resp. by specializing the parameter to a constant. We then use the "highest corner" of the specialized ideal to cut off high order terms from the polynomials during the standard basis computation over K to get the speed up. An important fact is that we can choose p as an arbitrary prime resp. as an arbitrary constant, not just a "lucky" resp. "random" one. Correctness of the algorithm will be deduced from a general semicontinuity theorem due to the first two authors. The computer algebra system Singular provides already the functionality to realize the algorithm and we present several examples illustrating its power.