Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields

  • Authors:

  • Erich Kaltofen;Pascal Koiran

  • Affiliations:

  • Massachusetts Institute of Technology, Cambridge, Massachusetts;École Normale Supérieure de Lyon, Lyon, France

  • Venue:
  • Proceedings of the 2006 international symposium on Symbolic and algebraic computation
  • Year:
  • 2006

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Abstract

We present algorithms that compute all irreducible factors of degree ≤ d of supersparse (lacunary) multivariate polynomials in n variables over an algebraic number field in deterministic polynomial-time in (l+d)n, where l is the size of the input polynomial. In supersparse polynomials, the term degrees enter logarithmically as their numbers of binary digits into the size measure l. The factors are again represented as supersparse polynomials. If the factors are represented as straight-line programs or black box polynomials, we can achieve randomized polynomial-time in (l+d)O(1). Our approach follows that by H. W. Lenstra, Jr., on computing factors of univariate supersparse polynomials over algebraic number fields. We generalize our ISSAC 2005 results for computing linear factors of supersparse bivariate polynomials over the rational numbers by appealing to recent lower bounds on the height of algebraic numbers and to a special case of the former Lang conjecture.