Global Newton Iteration over Archimedean and non-Archimedean Fields

TL;DR

This paper investigates global Newton iteration methods for algebraic sets over different fields, comparing approaches, proving equivalences, and analyzing computational efficiency for certifying approximate roots.

Contribution

It introduces and compares two Newton iteration approaches for RUR coefficients over various fields, providing explicit formulas and complexity analysis.

Findings

Over p-adic fields, both approaches yield the same iteration.

Over Archimedean fields, the approaches differ and explicit formulas are provided.

The methods can be computed efficiently and are useful for root certification.

Abstract

In this paper, we study iterative methods on the coefficients of the rational univariate representation (RUR) of a given algebraic set, called global Newton iteration. We compare two natural approaches to define locally quadratically convergent iterations: the first one involves Newton iteration applied to the approximate roots individually and then interpolation to find the RUR of these approximate roots; the second one considers the coefficients in the exact RUR as zeroes of a high dimensional map defined by polynomial reduction, and applies Newton iteration on this map. We prove that over fields with a p-adic valuation these two approaches give the same iteration function, but over fields equipped with the usual Archimedean absolute value, they are not equivalent. In the latter case, we give explicitly the iteration function for both approaches. Finally, we analyze the parallel complexity of the different versions of the global Newton iteration, compare them, and demonstrate that they can be efficiently computed. The motivation for this study comes from the certification of approximate roots of overdetermined and singular polynomial systems via the recovery of an exact RUR from approximate numerical data.