An Alternative Method for Primary Decomposition of Zero-dimensional Ideals over Finite Fields

TL;DR

This paper introduces a new method for primary decomposition of zero-dimensional ideals over finite fields, improving existing algorithms by leveraging Frobenius map invariants and enhancing polynomial factorization techniques.

Contribution

It proposes an alternative approach based on Frobenius map decomposition, enabling simultaneous computation of all primary components and improving polynomial factorization over finite fields.

Findings

Efficient computation of primary decomposition for zero-dimensional ideals.

Enhanced polynomial factorization method over finite fields.

Theoretical improvement of Berlekamp's algorithm.

Abstract

We present an alternative method for computing primary decomposition of zero-dimensional ideals over finite fields. Based upon the further decomposition of the invariant subspace of the Frobenius map acting on the quotient algebra in the algorithm given by S. Gao, D. Wan and M. Wang in 2008, we get an alternative approach to compute all the primary components at once. As one example of our method, an improvement of Berlekamp's algorithm by theoretical considerations which computes the factorization of univariate polynomials over finite fields is also obtained.