Computing sparse multiples of polynomials

Details Computing sparse multiples of polynomials Computing sparse multiples of polynomials

2010-09-16

arxiv.org/abs/1009.3214v2

Computer Science

Symbolic Computation

Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 6506

Scientific paper

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[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 a polynomial-time algorithm 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.

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