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Exceptional Polynomials over Arbitrary Fields

In: Algebra, Arithmetic and Geometry with Applications

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  • Robert M. Guralnick
  • Jan Saxl

Abstract

In 1993, Fried, Guralnick and Saxl classified indecomposable exceptional polynomials, which are not of affine type of degree a power of the characteristic, over finite fields (or more generally procyclic fields) of characteristics not 2 or 3 (and gave the group theoretic possibilities in characteristics 2 and 3). We give a different proof of this result which is valid over arbitrary fields. The proof is based on the classification of monodromy groups of indecomposable covers of curves with a totally ramified point obtained by the authors in earlier work. We also show that such polynomials are injective on rational points. We also discuss polynomials which are arithmetically indecomposable but geometrically decomposable.

Suggested Citation

  • Robert M. Guralnick & Jan Saxl, 2004. "Exceptional Polynomials over Arbitrary Fields," Springer Books, in: Chris Christensen & Avinash Sathaye & Ganesh Sundaram & Chandrajit Bajaj (ed.), Algebra, Arithmetic and Geometry with Applications, pages 457-472, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18487-1_25
    DOI: 10.1007/978-3-642-18487-1_25
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