Abstract. Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1, . . . , fm} ⊂ F[x1, . . . , xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper we design blackbox and efficient linear maps ϕ that reduce the number of variables from n to r but maintain trdeg{ϕ(fi)}i = r, assuming fi's sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing:1. Given a circuit C and sparse subcircuits f1, . (k, s, n) circuits. This partially generalizes the state of the art of depth-3 to depth-4 circuits. The notion of trdeg works best with large or zero characteristic, but we also give versions of our results for arbitrary fields.

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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 Fp (i.e. somehow avoid ∂x p /∂x = 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.

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