1,907 research outputs found

    Factoring multivariate polynomials over algebraic number fields

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    Factoring multivariate polynomials over algebraic number fields

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    Factoring multivariate polynomials over algebraic number fields

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    We present an algorithm to factor multivariate polynomials over algebraic number fields that is polynomial-time in the degrees of the polynomial to be factored. The algorithm is an immediate generalization of the polynomial-time algorithm to factor univariate polynomials with rational coefficients

    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

    Discovering the roots: Uniform closure results for algebraic classes under factoring

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    Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots rr is small but the multiplicities are exponentially large. Our method sets up a linear system in rr unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,,xn)f(x_1,\ldots,x_n) of size ss we prove that each factor has size at most a polynomial in: ss and the degree of the squarefree part of ff. Consequently, if f1f_1 is a 2Ω(n)2^{\Omega(n)}-hard polynomial then any nonzero multiple ifiei\prod_{i} f_i^{e_i} is equally hard for arbitrary positive eie_i's, assuming that ideg(fi)\sum_i \text{deg}(f_i) is at most 2O(n)2^{O(n)}. It is an old open question whether the class of poly(nn)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial ff of degree nO(1)n^{O(1)} and formula (resp. ABP) size nO(logn)n^{O(\log n)} we can find a similar size formula (resp. ABP) factor in randomized poly(nlognn^{\log n})-time. Consequently, if determinant requires nΩ(logn)n^{\Omega(\log n)} size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation τ\tau, f(τx)f(\tau\overline{x}) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

    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

    Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

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    In this paper we study the problem of deterministic factorization of sparse polynomials. We show that if fF[x1,x2,,xn]f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}] is a polynomial with ss monomials, with individual degrees of its variables bounded by dd, then ff can be deterministically factored in time spoly(d)logns^{\mathrm{poly}(d) \log n}. Prior to our work, the only efficient factoring algorithms known for this class of polynomials were randomized, and other than for the cases of d=1d=1 and d=2d=2, only exponential time deterministic factoring algorithms were known. A crucial ingredient in our proof is a quasi-polynomial sparsity bound for factors of sparse polynomials of bounded individual degree. In particular we show if ff is an ss-sparse polynomial in nn variables, with individual degrees of its variables bounded by dd, then the sparsity of each factor of ff is bounded by sO(d2logn)s^{O({d^2\log{n}})}. This is the first nontrivial bound on factor sparsity for d>2d>2. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Carath\'eodory's Theorem. Our work addresses and partially answers a question of von zur Gathen and Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the sparsity of factors of sparse polynomials

    Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach

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    We present a new algorithm for the computation of the irreducible factors of degree at most dd, with multiplicity, of multivariate lacunary polynomials over fields of characteristic zero. The algorithm reduces this computation to the computation of irreducible factors of degree at most dd of univariate lacunary polynomials and to the factorization of low-degree multivariate polynomials. The reduction runs in time polynomial in the size of the input polynomial and in dd. As a result, we obtain a new polynomial-time algorithm for the computation of low-degree factors, with multiplicity, of multivariate lacunary polynomials over number fields, but our method also gives partial results for other fields, such as the fields of pp-adic numbers or for absolute or approximate factorization for instance. The core of our reduction uses the Newton polygon of the input polynomial, and its validity is based on the Newton-Puiseux expansion of roots of bivariate polynomials. In particular, we bound the valuation of f(X,ϕ)f(X,\phi) where ff is a lacunary polynomial and ϕ\phi a Puiseux series whose vanishing polynomial has low degree.Comment: 22 page
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