The Method of Gauss-Newton to Compute Power Series Solutions of Polynomial Homotopies

TL;DR

This paper extends the Gauss-Newton method to compute power series solutions of polynomial homotopies, utilizing tropical algebraic geometry at singular points, and demonstrates its application with examples.

Contribution

It introduces a novel approach combining Gauss-Newton linearization with tropical algebraic geometry for power series solutions of polynomial homotopies.

Findings

Linearization leads to a block triangular system with cubic cost.

Regular case characterized by algebraic variety of an augmented system.

Singular points addressed using tropical algebraic geometry methods.

Abstract

We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.