Performance of Buchberger's Improved Algorithm using Prime Based Ordering

TL;DR

This paper evaluates prime-based ordering in Buchberger's improved algorithm for Groebner Bases, demonstrating significant computational time reductions through integer operations on power products encoded with primes.

Contribution

It introduces prime-based encoding of power products in Buchberger's algorithm, enabling faster integer operations and reducing computation time.

Findings

30% or more reduction in computation time on polynomial examples

Prime-based encoding simplifies power product operations

Integer-based implementations outperform string-based ones

Abstract

Prime-based ordering which is proved to be admissible, is the encoding of indeterminates in power-products with prime numbers and ordering them by using the natural number order. Using Eiffel, four versions of Buchberger's improved algorithm for obtaining Groebner Bases have been developed: two total degree versions, representing power products as strings and the other two as integers based on prime-based ordering. The versions are further distinguished by implementing coefficients as 64-bit integers and as multiple-precision integers. By using primebased power product coding, iterative or recursive operations on power products are replaced with integer operations. It is found that on a series of example polynomial sets, significant reductions in computation time of 30% or more are almost always obtained.