1,597 research outputs found

    Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

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    In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields

    Factoring bivariate sparse (lacunary) polynomials

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    We present a deterministic algorithm for computing all irreducible factors of degree ≤d\le d of a given bivariate polynomial f∈K[x,y]f\in K[x,y] over an algebraic number field KK and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in dd. Moreover, we show that the factors over \Qbarra of degree ≤d\le d which are not binomials can also be computed in time polynomial in the sparse length of the input and in dd.Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a multivariate version of Theorem 1 had independently been achieved by Erich Kaltofen and Pascal Koira

    Factoring bivariate lacunary polynomials without heights

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    We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap Theorem which allows to test whether a polynomial of the form P(X,X+1) is identically zero in time polynomial in the number of terms of P(X,Y). The algorithm we obtain 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. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.Comment: 25 pages, 1 appendi
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