The Proper Basis for Polynomial Ideals

TL;DR

The paper introduces the proper basis, a new ideal basis that enhances computational efficiency over Gr"obner and Buchberger's algorithms, especially for zero-dimensional polynomial ideals, by leveraging variable ordering and modular form.

Contribution

It defines and computes the proper basis, a novel approach that improves existing algorithms and reduces coefficient swell in polynomial ideal computations.

Findings

Proper basis is more efficient than Buchberger's and M"oller’s algorithms.

Proper basis reduces coefficient swell compared to Gr"obner basis.

Benchmark tests confirm the efficiency and modular advantages of the proper basis.

Abstract

We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The Gr\"obner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in $x_1$. The proper basis is defined and computed in the variables $\tilde{\bm{x}}:=(x_2,\dotsc,x_n)$ with $x_1$ serving as a parameter in the algebra $K[x_1][\tilde{\bm{x}}]$. Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of $\tilde{\bm{x}}$ unnecessarily involves the least variable $x_1$ but also M\"oller's algorithm due to its polynomial division mechanism. This is corroborated by a series of benchmark testings herein. The proper basis is in a modular form and neater than Gr\"obner basis and hence reduces its coefficient swell problem. It is expected that all the state of the art algorithms improving Buchberger's algorithm over the last decades can be further improved if we apply them to the proper basis.