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 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.

💡 Research Summary

The paper tackles a long‑standing gap in algebraic geometry and computational complexity: the lack of a Jacobian‑type criterion for testing algebraic independence of multivariate polynomials over fields of positive characteristic p. In characteristic 0 (or sufficiently large characteristic) the classical Jacobian matrix—whose entries are the ordinary partial derivatives—provides a necessary and sufficient condition: the polynomials are algebraically independent if and only if the Jacobian determinant is non‑zero. In characteristic p>0 this fails because the usual derivative annihilates p‑th powers (d(x^p)/dx = 0), so the Jacobian cannot detect dependencies that involve Frobenius powers.

The authors resolve this by moving from the ordinary differential calculus to the de Rham‑Witt complex, a construction introduced by Illusie in 1979 for crystalline cohomology. The de Rham‑Witt complex lives over the ring of p‑adic integers ℤ_p (more precisely over its unramified extensions) and carries a differential d_W that behaves like a “lifted” derivative: it respects the Witt vector structure and, crucially, does not kill p‑th powers. By lifting each input polynomial f_i ∈ F_p