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Multilinear Polynomial Systems: Root Isolation and Bit Complexity

Abstract : We exploit structure in polynomial system solving by considering polyno-mials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka generalized Sylvester matrices. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex. The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of Canny, Kaltofen, and Yagati [9] to the multi-homogeneous case. We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success 1 − 1/2 , for a given ≥ 1, and bit complexity O B (n 2 D 4+ (n N +1 + τ) + nD 2+ (D +)) for any > 0, where n is the number of variables, D equals the multilinear Bézout bound, N is the number of variable subsets, and τ is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input.
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Contributor : Angelos Mantzaflaris <>
Submitted on : Monday, April 15, 2019 - 10:24:05 AM
Last modification on : Friday, October 16, 2020 - 10:21:56 AM

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Ioannis Emiris, Angelos Mantzaflaris, Elias Tsigaridas. Multilinear Polynomial Systems: Root Isolation and Bit Complexity. Journal of Symbolic Computation, Elsevier, In press, Special Issue on Milestones in Computer Algebra (MICA 2016), ⟨10.1016/j.jsc.2020.06.005⟩. ⟨hal-02099556⟩

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