Gasull's Finite Moment Problem for Polynomials with Few Monomials

Let f ∈ C[x] have at most k monomials and put M_n = ∫_0^1 f(x)^n dx. A. Gasull asked whether there is a number N(k) such that M_1 = ⋯ = M_{N(k)} = 0 forces f = 0, and, if so, for its value or a good upper bound. We show that N(k) exists for every k. The proof treats the exponents as parameters, applies Hilbert's basis theorem to the moments with their denominators cleared, and uses the known fact, proved by Pakovich and by Françoise, Pakovich, Yomdin and Zhao, that the moments ∫_0^1 f^n dx, n ≥ 1, of a nonzero polynomial cannot all vanish. It gives no explicit bound. We also show that N(k) ≥ k for every k, that N(1) = 1 and N(2) = 2, and, by an exact computer-assisted certificate, that N(3) = 3: after a linear change of variables, a resultant that controls the case k = 3 has only nonnegative coefficients. The statement for k = 3 holds for all real exponents greater than −1/3. For every k, the first k moments force f = 0 when the exponents are sufficiently lacunary. For k = 4 and k = 5 we report exact certificates for all sets of integer exponents with largest exponent at most 50 and at most 16, respectively; these are finite-range computations. Numerically, the real-exponent version of the case k = 4 fails for some exponents with e_1 < 0. The value of N(k), and any explicit upper bound for it, remain open for k ≥ 4. This is an unrefereed note.