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TR12-014 | 20th February 2012 13:05

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



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 $\mathbb{F}_p$ (i.e. somehow avoid $\partial x^p/\partial x=0$) and thus capture algebraic independence.

We apply the new criterion to put the problem of testing algebraic independence in the complexity class $\mbox{NP}^\mbox{#P}$ (previously best was PSPACE). Also, we give a modest application to the problem of identity testing in algebraic complexity theory.

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