Root finding with threshold circuits

TL;DR

This paper demonstrates that for any fixed degree, complex roots of rational polynomials can be approximated efficiently using constant-depth threshold circuits, linking computational complexity with algebraic root finding.

Contribution

It introduces a uniform TC^0 algorithm for approximating roots of fixed-degree polynomials from their coefficients, using power series inversion.

Findings

Roots of fixed-degree polynomials are computable in TC^0.

The method employs power series to invert polynomials efficiently.

Application to bounded arithmetic theory VTC^0 is discussed.

Abstract

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.