A Kronecker algorithm for locally closed sets over a perfect field

TL;DR

This paper introduces a probabilistic Kronecker algorithm for computing representations of zero-dimensional sections of algebraic varieties over perfect fields, combining deformation, Newton-Hensel lifting, and elimination techniques.

Contribution

It presents a novel probabilistic algorithm with quadratic complexity for representing zero-dimensional sections of varieties over perfect fields, improving computational efficiency.

Findings

Algorithm achieves soft-quadratic complexity in input size.

Provides complexity bounds for finite fields and rational numbers.

Utilizes homotopic deformation and Newton-Hensel lifting methods.

Abstract

We develop a probabilistic algorithm of Kronecker type for computing a Kronecker representation of a zero-dimensional linear section of an algebraic variety $V$ defined over a perfect field $k$. The variety $V$ is the Zariski closure of the set of common zeros $\{F_1=0,\ldots,F_r=0,G\not=0\}$ of multivariate polynomials $F_1,\ldots,F_r\in k[X_1,\ldots,X_n]$ outside a prescribed hypersurface $\{G=0\}$. We assume that $F_1,\ldots,F_r$ satisfy natural geometric conditions, such as regularity and radicality, in the local ring $k[X_1,\ldots,X_n]_G$. Our approach combines homotopic deformation techniques with symbolic Newton-Hensel lifting and elimination. We discuss the concept of lifting curves as intermediate geometric objects that enable efficient computation. The complexity of the algorithm is expressed in terms of the degrees and arithmetic size of the input and achieves soft-quadratic complexity in these parameters. We provide detailed complexity analyses for arbitrary perfect fields, as well as for two important cases in computer algebra: finite fields and the field of rational numbers. For each case, we obtain sharp bounds on the size of the base field or required primes.