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Constructing Faithful Homomorphisms over Fields of Finite Characteristic

Published:26 June 2023Publication History
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Abstract

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [3] and exploited by them, and Agrawal et al. [2] to design algebraic independence–based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields.

Building on a recent criterion of Pandey et al. [15], we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. [3] and Agrawal et al. [2] in the positive characteristic setting.

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 15, Issue 1-2
      June 2023
      58 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/3605363
      Issue’s Table of Contents

      Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 26 June 2023
      • Accepted: 12 January 2023
      • Received: 12 May 2021
      Published in toct Volume 15, Issue 1-2

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