Separation bounds for polynomial systems

Abstract : Based on recent advances on aggregate separation bounds for univariate polynomials, we extend these results to the isolated roots of zero-dimensional, as well as positive-dimensional and overdetermined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume. This improves Canny's gap theorem [8]. Our results exploit sparseness; they are in general comparable to the lower bounds on the absolute value of the root coordinates by Brownawell and Yap [6] which, however, were obtained under a strong hypothesis, namely the existence of a zero-dimensional projection. We also present polynomial systems whose roots almost attain our zero bounds. Our bounds are applied to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix, thus showing the problem is of polynomial bit complexity, in order to bound the value of a positive polynomial over the simplex, thus improving by at least one order of magnitude upon all existing bounds, and in order to asymptotically bound the number of steps of any subdivision-based algorithm that isolates all real roots of a polynomial system.
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https://hal.inria.fr/hal-01105276
Contributor : Elias Tsigaridas <>
Submitted on : Wednesday, January 21, 2015 - 10:47:11 PM
Last modification on : Wednesday, October 17, 2018 - 5:02:04 PM
Long-term archiving on: Wednesday, April 22, 2015 - 10:23:56 AM

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  • HAL Id : hal-01105276, version 1

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Ioannis Emiris, Bernard Mourrain, Elias Tsigaridas. Separation bounds for polynomial systems. 2015. ⟨hal-01105276v1⟩

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