Algebraic independence in positive characteristic: A $p$-adic calculus

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, no analogous characterization is 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 functorial de Rham-Witt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural explicit generalization of the Jacobian. We call this new avatar the Witt-Jacobian. In essence, we show how to faithfully differentiate polynomials over Fp (i.e., somehow avoid ∂xp/∂x = 0) and thus capture algebraic independence. We give two applications of this criterion in algebraic complexity theory.