34 research outputs found

    On fully split lacunary polynomials in finite fields

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    We estimate the number of possible types degree patterns of kk-lacunary polynomials of degree t<pt < p which split completely modulo pp. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.Comment: 8 pages. Bull. Polish Acad. Sci. Math., to appea

    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 fK[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

    Bounded-degree factors of lacunary multivariate polynomials

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    In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization. The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable.Comment: 31 pages; Long version of arXiv:1401.4720 with simplified proof

    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

    Finite Fields: Theory and Applications

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    Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

    The Multivariate Resultant is NP-hard in any Characteristic

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    The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NP-hardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our 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). We also observe that in characteristic zero, this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE.Comment: 13 page

    New Bounds on Quotient Polynomials with Applications to Exact Divisibility and Divisibility Testing of Sparse Polynomials

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    A sparse polynomial (also called a lacunary polynomial) is a polynomial that has relatively few terms compared to its degree. The sparse-representation of a polynomial represents the polynomial as a list of its non-zero terms (coefficient-degree pairs). In particular, the degree of a sparse polynomial can be exponential in the sparse-representation size. We prove that for monic polynomials f,gC[x]f, g \in \mathbb{C}[x] such that gg divides ff, the 2\ell_2-norm of the quotient polynomial f/gf/g is bounded by f1O~(g03deg2f)g01\lVert f \rVert_1 \cdot \tilde{O}(\lVert{g}\rVert_0^3\text{deg}^2{ f})^{\lVert{g}\rVert_0 - 1}. This improves upon the exponential (in degf\text{deg}{ f}) bounds for general polynomials and implies that the trivial long division algorithm runs in time quasi-linear in the input size and number of terms of the quotient polynomial f/gf/g, thus solving a long-standing problem on exact divisibility of sparse polynomials. We also study the problem of bounding the number of terms of f/gf/g in some special cases. When f,gZ[x]f, g \in \mathbb{Z}[x] and gg is a cyclotomic-free (i.e., it has no cyclotomic factors) trinomial, we prove that f/g0O(f0size(f)2log6degg)\lVert{f/g}\rVert_0 \leq O(\lVert{f}\rVert_0 \text{size}({f})^2 \cdot \log^6{\text{deg}{ g}}). When gg is a binomial with g(±1)0g(\pm 1) \neq 0, we prove that the sparsity is at most O(f0(logf0+logf))O(\lVert{f}\rVert_0 ( \log{\lVert{f}\rVert_0} + \log{\lVert{f}\rVert_{\infty}})). Both upper bounds are polynomial in the input-size. We leverage these results and give a polynomial time algorithm for deciding whether a cyclotomic-free trinomial divides a sparse polynomial over the integers. As our last result, we present a polynomial time algorithm for testing divisibility by pentanomials over small finite fields when degf=O~(degg)\text{deg}{ f} = \tilde{O}(\text{deg}{ g})

    Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

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    We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PP^{NP^NP}, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract which appeared in STOC 1999. This version includes significant corrections and improvements to various asymptotic bounds. Needs cjour.cls to compil

    Deterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials

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    We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [von zur Gathen/Kaltofen, J. Comp. Sys. Sci., 1985] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal by introducing essential factorization schemes which can be thought of as a relaxation of the regular factorization notion
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