On the complexity of Chow and Hurwitz forms

TL;DR

This paper introduces a deterministic algorithm for computing Chow and Hurwitz forms with a proven single exponential bit complexity bound, advancing computational algebra in incidence geometry.

Contribution

It presents the first bit complexity bound for Chow forms and extends the algorithm to Hurwitz forms, connecting them with matroid theory.

Findings

Developed a deterministic algorithm using resultants

Established a single exponential complexity upper bound

Extended the approach to Hurwitz forms in projective space

Abstract

We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model, and our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space, and explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.