The Proper Basis for Polynomial Ideals

Abstract

We define a new type of ideal basis called the proper basis that improves both Gröbner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dots ,x_n]$ over a field $K$. The Gröbner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in $x_1$. The proper basis is defined and computed in the variables $\tilde{\mathbf{x}}:=(x_2,\dots ,x_n)$ with $x_1$ serving as a parameter in the algebra $K[x_1][\tilde{\mathbf{x}}]$. Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of $\tilde{\mathbf{x}}$ unnecessarily involves the least variable $x_1$ but also Möller'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 Gröbner 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.

15 pages. The length of the previous version is shortened to 15 pages in its current form