Algebraic independence in positive characteristic: A ๐-adic calculus
Johannes Mittmann, Nitin Saxena, Peter Scheiblechner ยท Transactions of the American Mathematical Society ยท 2014
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 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 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 F p \mathbb {F}_p (i.e., somehow avoid โ x p / โ x = 0 \partial x^p/\partial x=0 ) and thus capture algebraic independence. We give two applications of this criterion in algebraic complexity theory.