Overdetermined systems of sparse polynomial equations

Abstract : We show that, for a system of univariate polynomials given in the sparse encoding, we can compute a single polynomial defining the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine if such a system of polynomials does have a zero in time quasi-linear in the logarithm of the degree. The underlying algorithm relies on a result of Bombieri and Zannier on multiplicatively dependent points in subvarieties of an algebraic torus. We also present the following conditional partial extension to the higher dimensional setting. Assume that the effective Zilber conjecture holds true. Then, for a system of multivariate polynomials given in the sparse encoding, we can compute a finite collection of complete intersections outside a hypersurface that defines the same zero set, in time quasi-linear in the logarithm of the degree.
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Francesco Amoroso, Louis Leroux, Martin Sombra. Overdetermined systems of sparse polynomial equations. Foundations of Computational Mathematics, Springer Verlag, 2014, 15 (1), pp.53-87. ⟨hal-00677676v4⟩

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