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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Algebraic independence in positive characteristic: A $p$-adic calculus
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by Johannes Mittmann, Nitin Saxena and Peter Scheiblechner PDF
Trans. Amer. Math. Soc. 366 (2014), 3425-3450 Request permission

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
  • Received by editor(s): May 14, 2012
  • Published electronically: March 14, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3425-3450
  • MSC (2010): Primary 12Y05, 13N05, 14F30, 03D15, 68Q17, 68W30
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06268-5
  • MathSciNet review: 3192602