No abstract

We consider the problem of finding a sparse multiple of a polynomial. Given f ∈ 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 ∈ 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.

No abstract

We investigate the problem of computing the sparsest multiple of a given univariate polynomial. Over the rational numbers we exhibit algorithms for when the target sparsity is two (i.e., the existence of a binomial multiple), as well as some cases when the target multiple has greater (constant) sparsity. Over finite fields, we tie the cost of finding a binomial multiple (of low degree) to order-finding in the multiplicative group of a finite field, as well as presenting similar, but more limited results for general sparsit.