Solving bihomogeneous polynomial systems with a zero-dimensional projection

TL;DR

This paper introduces a novel linear algebra-based method to compute zero-dimensional projections of bihomogeneous polynomial systems, extending classical algorithms to handle non-zero dimensional varieties efficiently.

Contribution

It develops new multiplication maps for bihomogeneous systems, generalizes the FGLM algorithm, and provides complexity bounds for computing these maps and Gr"obner bases.

Findings

Eigenvalues of multiplication maps enable numerical approximation of projections.

The method extends classical zero-dimensional algorithms to non-zero dimensional cases.

A single exponential complexity bound is established based on bidegrees.

Abstract

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Gr\"obner bases. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Gr\"obner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal.