Abstract
In this paper, we show that for a system of univariate polynomials given in sparse encoding we can compute a single polynomial defining the same zero set in quasilinear time in the logarithm of the degree. In particular, it is possible to decide whether such a system of polynomials has a zero in quasilinear time 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. Then, for a system of multivariate polynomials given in sparse encoding we can compute a finite collection of complete intersections outside hypersurfaces that defines the same zero set in quasilinear time in the logarithm of the degree.
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Acknowledgments
We thank Gaël Rémond and Umberto Zannier for useful discussions about unlikely intersections and the Zilber conjecture. We also thank the referees for their many remarks that helped us to improve our presentation. Part of this work was done while the authors met at the Universitat de Barcelona, the Université de Caen, and the Centro di Ricerca Matematica Ennio de Giorgi (Pisa). We thank these institutions for their hospitality. Amoroso and Leroux were partially supported by the International Programs for Scientific Cooperation (PICS) of the Centre National de la Recherche Scientifique (CNRS), “Properties of the heights of arithmetic varieties.” Amoroso was also partially supported by the ANR research project “Hauteurs, modularité, transcendance.” Sombra was partially supported by the MICINN research projects MTM2009-14163-C02-01 and MTM2012-38122-C03-02.
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Communicated by Felipe Cucker.
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Amoroso, F., Leroux, L. & Sombra, M. Overdetermined Systems of Sparse Polynomial Equations. Found Comput Math 15, 53–87 (2015). https://doi.org/10.1007/s10208-014-9207-y
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DOI: https://doi.org/10.1007/s10208-014-9207-y