Algebraic Independence in Positive Characteristic -- A p-Adic Calculus

TL;DR

This paper introduces a novel criterion called the Witt-Jacobian for testing algebraic independence of polynomials over fields of positive characteristic, extending the Jacobian criterion to characteristic p>0 using p-adic and de Rham-Witt complex techniques.

Contribution

It provides the first algebraic independence criterion in positive characteristic fields based on p-adic calculus and the de Rham-Witt complex, generalizing the Jacobian criterion.

Findings

The criterion is based on a non-degeneracy condition on a lift of the Jacobian polynomial.

It enables testing algebraic independence in NP^#P complexity class.

Application to identity testing in algebraic complexity theory.

Abstract

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic p>0, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to a non-degeneracy condition on a lift of the Jacobian polynomial over (an unramified extension of) the ring of p-adic integers. Our proof builds on the de Rham-Witt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural generalization of the Jacobian. This new avatar we call the Witt-Jacobian. In essence, we show how to faithfully differentiate polynomials over F_p (i.e. somehow avoid dx^p/dx=0) and thus capture algebraic independence. We apply the new criterion to put the problem of testing algebraic independence in the complexity class NP^#P (previously best was PSPACE). Also, we give a modest application to the problem of identity testing in algebraic complexity theory.