Algorithms for Computing Sparse Shifts for Multivariate Polynomials

In this paper, we investigate the problem of finding t-sparse shifts for multivariate polynomials. Given a polynomial f∈ℱ[x1, x2, …, xn] of degree d, and a positive integer t, we consider the problem of representing f(x) as a ?-linear combination of the power products of ui where ui = xi−bi for some bi∈?, an extension of ℱ, for i = 1, …, n, i.e., f = ∑jFjuαj, in which at most t of the Fj are non-zero. We provide sufficient conditions for uniqueness of sparse shifts for multivariate polynomials, prove tight bounds on the degree of the polynomial being interpolated in terms of the sparsity bound t and a bound on the size of the coefficients of the polynomial in the standard representation, and describe two new efficient algorithms for computing sparse shifts for a multivariate polynomial.