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Randomized Root Finding over Finite FFT-fields using Tangent Graeffe Transforms

This work studies the problem of computing all roots of a polynomial that splits over Fq, which was one of the bottlenecks for fast sparse interpolation in practice, and revisits and slightly improves existing algorithms and then presents new randomized ones based on the Graeffe transform.
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The Multivariate Resultant Is NP-hard in Any Characteristic

The main result is that testing the resultant for zero is NP-hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension).
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AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM

The Permanent versus Determinant problem is the following: Given an n × n matrix X of indeterminates over a field of characteristic different from two, find the smallest matrix M whose coefficients…

Fast in-place algorithms for polynomial operations: division, evaluation, interpolation

New in-place algorithms for the aforementioned polynomial computations which require only constant extra space and achieve the same asymptotic running time as their out-of-place counterparts are demonstrated.
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The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

It is shown that the real tau-conjecture holds true for a restricted class of sums of products of sparse polynomials, and lower bounds for arestricted class of depth-4 circuits are shown.
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On the complexity of the multivariate resultant

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Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

It is shown that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses, and algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits are deployed.
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Deterministic root finding over finite fields using Graeffe transforms

New deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field, and a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions.
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Bounded-degree factors of lacunary multivariate polynomials

  • Bruno Grenet
  • Mathematics, Computer Science
    Journal of symbolic computation
  • 11 December 2014
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Factoring bivariate lacunary polynomials without heights

The algorithm obtained is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients, and can be used to find some linear factors of bivariate lacunary polynomsials over a field of large finite characteristic in probabilistic polynomial time.
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