Binomiality Testing and Computing Sparse Polynomials via Witness Sets

Sparse polynomials that vanish on algebraic sets are preferred in many computations since they are easy to evaluate and often arise from underlying structure. For example, a monomial vanishes on an algebraic set if and only if the algebraic set is contained in the union of the coordinate hyperplanes. Eisenbud and Sturmfels initiated a detailed study of binomial ideals roughly 25 years ago and showed that they had many special properties including that each component is rational. Given a general point on a component and its tangent space, this paper exploits rationality to develop a local approach that decides if the component is defined by binomials or not. When a component is not defined by binomials, one often is interested in computing sparse polynomials that vanish on the component. Thus, this paper also develops an approach for computing sparse polynomials using a witness set for the component. Our approach relies on using numerical homotopy methods to sample points on the algebraic set along with incorporating multiplicity information using Macaulay dual spaces. If the algebraic set is defined by polynomials with rational coefficients, exactness recovery such as lattice based methods can be used to find exact representations of the sparse polynomials. Several examples are presented demonstrating the new methods.