Abstract
We consider the problem of finding a sparse multiple of a polynomial. Given f ∈ F[x] of degree d, and a desired sparsity t, our goal is to determine if there exists a multiple h ∈ 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.
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), and MITACS.
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Giesbrecht, M., Roche, D.S., Tilak, H. (2010). Computing Sparse Multiples of Polynomials. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17517-6_25
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