664 research outputs found

    Proving formally the implementation of an efficient gcd algorithm for polynomials

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    Last version published in the proceedings of IJCAR 06, part of FLOC 06.International audienceWe describe here a formal proof in the Coq system of the structure theorem for subresultants, which allows to prove formally the correction of our implementation of the subresultant algorithm. Up to our knowledge, it is the first mechanized proof of this result

    Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

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    The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time O ((log N)^5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4). The aim of this article is to describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits

    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

    Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

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    We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., (xnQ)(x^n \in \mathbb{Q}) and (xnZ)(x^n \in \mathbb{Z}). Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated Deduction, 2015. Proceedings to be published by Springer-Verla

    Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

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    We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial
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