, Volume 64, Issue 3, pp 454–480 | Cite as

Computing Sparse Multiples of Polynomials

  • Mark GiesbrechtEmail author
  • Daniel S. Roche
  • Hrushikesh Tilak


We consider the problem of finding a sparse multiple of a polynomial. Given fF[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple hF[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=ℚ and t is constant, we give an algorithm which requires polynomial-time in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.


Sparse polynomial Sparsest multiple 



The authors would like to thank John May, Arne Storjohann, and the anonymous referees for their careful reading and useful observations on earlier versions of this work.


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Copyright information

© Springer Science+Business Media, LLC (outside the USA) 2012

Authors and Affiliations

  • Mark Giesbrecht
    • 1
    Email author
  • Daniel S. Roche
    • 2
  • Hrushikesh Tilak
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.United States Naval AcademyAnnapolisUSA

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