6,259 research outputs found

    Multivariate sparse interpolation using randomized Kronecker substitutions

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    We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial, that is, its sparsity. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm.Comment: 21 pages, 2 tables, 1 procedure. Accepted to ISSAC 201

    Reconstructing Rational Functions with FireFly\texttt{FireFly}

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    We present the open-source C++\texttt{C++} library FireFly\texttt{FireFly} for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use FireFly\texttt{FireFly} in the context of integration-by-parts reductions and compare runtime and memory consumption to a fully algebraic approach with the program Kira\texttt{Kira}.Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio

    FORM version 4.0

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    We present version 4.0 of the symbolic manipulation system FORM. The most important new features are manipulation of rational polynomials and the factorization of expressions. Many other new functions and commands are also added; some of them are very general, while others are designed for building specific high level packages, such as one for Groebner bases. New is also the checkpoint facility, that allows for periodic backups during long calculations. Lastly, FORM 4.0 has become available as open source under the GNU General Public License version 3.Comment: 26 pages. Uses axodra

    How proofs are prepared at Camelot

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    We study a design framework for robust, independently verifiable, and workload-balanced distributed algorithms working on a common input. An algorithm based on the framework is essentially a distributed encoding procedure for a Reed--Solomon code, which enables (a) robustness against byzantine failures with intrinsic error-correction and identification of failed nodes, and (b) independent randomized verification to check the entire computation for correctness, which takes essentially no more resources than each node individually contributes to the computation. The framework builds on recent Merlin--Arthur proofs of batch evaluation of Williams~[{\em Electron.\ Colloq.\ Comput.\ Complexity}, Report TR16-002, January 2016] with the observation that {\em Merlin's magic is not needed} for batch evaluation---mere Knights can prepare the proof, in parallel, and with intrinsic error-correction. The contribution of this paper is to show that in many cases the verifiable batch evaluation framework admits algorithms that match in total resource consumption the best known sequential algorithm for solving the problem. As our main result, we show that the kk-cliques in an nn-vertex graph can be counted {\em and} verified in per-node O(n(ω+ϵ)k/6)O(n^{(\omega+\epsilon)k/6}) time and space on O(n(ω+ϵ)k/6)O(n^{(\omega+\epsilon)k/6}) compute nodes, for any constant ϵ>0\epsilon>0 and positive integer kk divisible by 66, where 2ω<2.37286392\leq\omega<2.3728639 is the exponent of matrix multiplication. This matches in total running time the best known sequential algorithm, due to Ne{\v{s}}et{\v{r}}il and Poljak [{\em Comment.~Math.~Univ.~Carolin.}~26 (1985) 415--419], and considerably improves its space usage and parallelizability. Further results include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph.Comment: 42 p

    Modular Las Vegas Algorithms for Polynomial Absolute Factorization

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    Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of ff, or more precisely, of ff modulo some prime integer pp. The same idea of choosing a pp satisfying some prescribed properties together with LLLLLL is used to provide a new strategy for absolute factorization of f(X,Y)f(X,Y). We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400

    Solving Degenerate Sparse Polynomial Systems Faster

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    Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve
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