The number of decomposable univariate polynomials. extended abstract

  • Authors:

  • Joachim von zur Gathen

  • Affiliations:

  • Universität Bonn, Bonn, Germany

  • Venue:
  • Proceedings of the 2009 international symposium on Symbolic and algebraic computation
  • Year:
  • 2009

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Abstract

A univariate polynomial f over a field is decomposable if it is the composition f = g ⊕ h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposable polynomials over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker. A centerpiece of our approach is a decomposition algorithm in the wild case, which shows that sufficiently many polynomials are decomposable.