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


Authors: Johannes Mittmann, Nitin Saxena and Peter Scheiblechner
Journal: Trans. Amer. Math. Soc. 366 (2014), 3425-3450
MSC (2010): Primary 12Y05, 13N05, 14F30, 03D15, 68Q17, 68W30
Published electronically: March 14, 2014
MathSciNet review: 3192602
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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$, 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 $ \mathbb{F}_p$ (i.e., somehow avoid $ \partial x^p/\partial x=0$) and thus capture algebraic independence.

We give two applications of this criterion in algebraic complexity theory.


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Additional Information

Johannes Mittmann
Affiliation: Hausdorff Center for Mathematics, Endenicher Allee 62, D-53115 Bonn, Germany
Email: johannes.mittmann@hcm.uni-bonn.de

Nitin Saxena
Affiliation: Department of Computer Science & Engineering, IIT Kanpur, 208016 Kanpur, India
Email: nitin@cse.iitk.ac.in

Peter Scheiblechner
Affiliation: Hochschule Luzern - Technik & Architektur, Technikumstrasse 21, CH-6048 Horw, Switzerland
Email: peter.scheiblechner@hslu.ch

DOI: https://doi.org/10.1090/S0002-9947-2014-06268-5
Keywords: Algebraic independence, crystalline cohomology, de Rham, differential, finite field, Galois ring, identity testing, Jacobian, K\"ahler, $p$-adic, Teichm\"uller, Witt, zeta function
Received by editor(s): May 14, 2012
Published electronically: March 14, 2014
Article copyright: © Copyright 2014 American Mathematical Society