A New Type of Gr\"obner Basis and Its Complexity

TL;DR

This paper introduces a new type of Gr"obner basis that reduces computational complexity and expression swell, combining advantages of Buchberger's algorithm and Wu's method for zero-dimensional polynomial ideals.

Contribution

It proposes a novel ideal basis that balances complexity reduction with retention of algebraic information, improving efficiency over traditional methods.

Findings

Requires fewer word operations than Buchberger's algorithm

Reduces intermediate expression swell significantly

Retains algebraic information and solves the ideal membership problem

Abstract

The new type of ideal basis introduced herein constitutes a compromise between the Gr\"obner bases based on the Buchberger's algorithm and the characteristic sets based on the Wu's method. It reduces the complexity of the traditional Gr\"obner bases and subdues the notorious intermediate expression swell problem and intermediate coefficient swell problem to a substantial extent. The computation of an $S$-polynomial for the new bases requires at most $O(m\ln^2m\ln\ln m)$ word operations whereas $O(m^6\ln^2m)$ word operations are requisite in the Buchberger's algorithm. Here $m$ denotes the upper bound for the numbers of terms both in the leading coefficients and for the rest of the polynomials. The new bases are for zero-dimensional polynomial ideals and based on univariate pseudo-divisions. However in contrast to the pseudo-divisions in the Wu's method for the characteristic sets, the new bases retain the algebraic information of the original ideal and in particular, solve the ideal membership problem. In order to determine the authentic factors of the eliminant, we analyze the multipliers of the pseudo-divisions and develop an algorithm over principal quotient rings with zero divisors.