Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation最新文献

{"title":"The Frobenius FFT","authors":"J. Hoeven, Robin Larrieu","doi":"10.1145/3087604.3087633","DOIUrl":"https://doi.org/10.1145/3087604.3087633","url":null,"abstract":"Let Fq be the finite field with q elements and let ω be a primitive n-th root of unity in an extension field Fqd of Fq. Given a polynomial P ∈ Fq [x] of degree less than n, we will show that its discrete Fourier transform (P (ω0), ..., P (ωn - 1)) ∈ Fqdn can be computed essentially d times faster than the discrete Fourier transform of a polynomial Q ∈ Fqd [x] of degree less than n, in many cases. This result is achieved by exploiting the symmetries provided by the Frobenius automorphism of Fqd over Fq.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115467593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exploring the Dynamic Buchberger Algorithm","authors":"J. Perry","doi":"10.1145/3087604.3087643","DOIUrl":"https://doi.org/10.1145/3087604.3087643","url":null,"abstract":"Abstract \"Static\" Buchberger algorithms to compute a Gröbner basis require as input both a set of polynomials and a term ordering. \"Dynamic\" Buchberger algorithms do not require an ordering as input, but compute instead a Gröbner basis with respect to a \"discovered\" ordering. A good heuristic usually results in a basis that is relatively small, a desirable property for many applications. This article uses a new C++ implementation to explore variants of the original algorithm. While the implementation is preliminary, and in general much slower than well-known static implementations, it still succeeds at computing some bases more quickly.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123310492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gcd Modulo a Primary Triangular Set of Dimension Zero","authors":"X. Dahan","doi":"10.1145/3087604.3087612","DOIUrl":"https://doi.org/10.1145/3087604.3087612","url":null,"abstract":"Computing gcd over a triangular set T is the core routine of the machinery of some triangular decomposition methods, in the realm of polynomial ideal theory. As such it has been studied intensively and is well-understood and implemented in several situations, especially in the case where coefficients are over a radical triangular set; It is not the case over a non-radical one. This paper introduces a gcd notion in this case, when additionally for simplicity T is assumed to be primary. It is built upon the Henselian property of the coefficient ring, and is natural in that it is linked with the subresultant sequence of a and b modulo T. A general algorithm still relies on some assumptions, except for the case of a triangular set of one variable.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122427042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Big Prime Field FFT on the GPU","authors":"Liangyu Chen, S. Covanov, Davood Mohajerani, M. M. Maza","doi":"10.1145/3087604.3087657","DOIUrl":"https://doi.org/10.1145/3087604.3087657","url":null,"abstract":"We consider prime fields of large characteristic, typically fitting on $k$ machine words, where k is a power of 2. When the characteristic of these fields is restricted to a subclass of the generalized Fermat numbers, we show that arithmetic operations in such fields offer attractive performance, both in terms of algebraic complexity and parallelism. In particular, these operations can be vectorized, leading to efficient implementation of fast Fourier transforms on graphics processing units.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121322293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Extended Hensel Construction and its Application to the Computation of Limit Points","authors":"P. Alvandi, M. Ataei, M. M. Maza","doi":"10.1145/3087604.3087658","DOIUrl":"https://doi.org/10.1145/3087604.3087658","url":null,"abstract":"The Extended Hensel Construction (EHC) is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. This work is motivated and illustrated by the computation of real branches of space curves.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132022532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Linear Time Fourier Transforms of Sn-k-invariant Functions on the Symmetric Group Sn","authors":"M. Clausen, Paul Hühne","doi":"10.1145/3087604.3087628","DOIUrl":"https://doi.org/10.1145/3087604.3087628","url":null,"abstract":"This paper introduces new techniques for the efficient computation of discrete Fourier transforms (DFTs) of Sn-k-invariant functions on the symmetric group Sn. We uncover diamond- and leaf-rake-like structures in Young's seminormal and orthogonal representations. Combining this with both a multiresolution scheme and an anticipation technique for saving scalar multiplications leads to linear time partial FFTs. Following the inductive version of Young's branching rule we obtain a global FFT that computes a DFT of Sn-k-invariant functions on Sn in at most ck...[Sn : Sn-k] scalar multiplications and additions, where ck denotes a positive constant depending only on k. This run-time, which is linear in [Sn : Sn-k], is order optimal and improves Maslen's algorithm. For example, it takes less than one second on a standard notebook to run our FFT algorithm for an Sn-2-invariant real-valued function on Sn, n=5000.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"172 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115131130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing with Arithmetic Groups","authors":"G. Nebe","doi":"10.1145/3087604.3087661","DOIUrl":"https://doi.org/10.1145/3087604.3087661","url":null,"abstract":"Around 1900 Voronoï [10] formulated his fundamental algorithm to enumerate all similarity classes of perfect lattices in a given dimension. This algorithm has far reaching generalisations ([7], [2], [6], [4]) used to compute generators and relators but also the integral homology for arithmetic groups. A quite general situation, where one may apply Voronoï’s algorithm, is described in [7]: Let σ : V1 ×V2 → R be a non-degenerate bilinear mapping on a pair of isomorphic finite-dimensional real vector spaces V1, V2. Two open non-empty subsets Pi ⊆ Vi form a pair of dual cones, if σ is strictly positive on P1 × P2 and for all f ∈ V1 P1, y ∈ V2 P2 there are f ′ ∈ P1, y′ ∈ P2 such that σ ( f ,y′) ≤ 0, σ ( f ′,y) ≤ 0. We now fix a discrete subset D ⊆ P2 {0}. For f ∈ P1 we define • The minimum of f as min( f ) := min{σ ( f ,d ) | d ∈ D}, • S ( f ) := {d ∈ D | σ ( f ,d ) = min( f )}, • and the Voronoï domain V ( f ) := {∑d ∈S (f ) add | ad ≥ 0}. • The element f is called perfect, if S ( f ) spans V2, so if V ( f ) has a non-empty interior. • PD := { f ∈ P1 | min( f ) = 1, f is perfect } denotes the set of perfect elements of minimum 1.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122134277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Resultants and Discriminants for Bivariate Tensor-Product Polynomials","authors":"Angelos Mantzaflaris, Elias P. Tsigaridas","doi":"10.1145/3087604.3087646","DOIUrl":"https://doi.org/10.1145/3087604.3087646","url":null,"abstract":"Optimal resultant formulas have been systematically constructed mostly for unmixed polynomial systems, that is, systems of polynomials which all have the same support. However, such a condition is restrictive, since mixed systems of equations arise frequently in practical problems. We present a square, Koszul-type matrix expressing the resultant of arbitrary (mixed) bivariate tensor-product systems. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the entries of the matrix are simply coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. Moreover, for tensor-product systems with more than two (affine) variables, we prove an impossibility result: no universal degree-one formulas are possible, unless the system is unmixed. We present applications of the new construction in the computation of discriminants and mixed discriminants as well as in solving systems of bivariate polynomials with tensor-product structure.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129975205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"M4GB: An Efficient Gröbner-Basis Algorithm","authors":"R. Makarim, M.M.J. Stevens","doi":"10.1145/3087604.3087638","DOIUrl":"https://doi.org/10.1145/3087604.3087638","url":null,"abstract":"This paper introduces a new efficient algorithm for computing Grobner-bases named M4GB. Like Faugere's algorithm F4 it is an extension of Buchberger's algorithm that describes: how to store already computed (tail-)reduced multiples of basis polynomials to prevent redundant work in the reduction step; and how to exploit efficient linear algebra for the reduction step. In comparison to F4 it removes further redundant work in the processing of reducible monomials. Furthermore, instead of translating the reduction of many critical pairs into the row reduction of some large matrix, our algorithm is described more natively and is efficient while processing critical pairs one by one. This feature implies that typically M4GB has to process fewer critical pairs than F4, and reduces the time and data complexity 'staircase' related to the increasing degree of regularity for a sequence of problems one observes for F4. To achieve high efficiency, M4GB has been designed specifically to operate only on tail-reduced polynomials, i.e., polynomials of which all terms except the leading term are non-reducible. This allows it to perform full-reduction directly in the computation of a term polynomial multiplication, where all computations are done over coefficient vectors over the non-reducible monomials. We have implemented a version of our new algorithm tailored for dense overdefined polynomial systems as a proof of concept and made our source code publicly available. We have made a comparison of our implementation against the implementations of FGBlib, Magma and OpenF4 on various dense Fukuoka MQ challenge problems that we were able to compute in reasonable time and memory. We observed that M4GB uses the least total CPU time and the least memory of all these implementations for those MQ problems, often by a significant factor. In the Fukuoka MQ challenges, the starting challenges of Type V and Type VI have 16 equations which was chosen based on an extrapolated computational runtime of more than a month using Magma. M4GB allowed us to set new records for these Fukuoka MQ challenges breaking Type V (F28) up to 18 equations and Type VI (F31) up to 19 equations, each can be computed within up to 11 days on our dual Xeon system.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127626578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved Bounds on Absolute Positiveness of Multivariate Polynomials","authors":"Swaroop N. Prabhakar, Vikram Sharma","doi":"10.1145/3087604.3087631","DOIUrl":"https://doi.org/10.1145/3087604.3087631","url":null,"abstract":"A multivariate polynomial F(x1,x2,...,xn) is said to be absolutely positive from a real number B if F and all its partial derivatives are non-negative for x1,x2,...,xn ≥ B. One of the well known bounds on absolute positiveness in the literature is due to Hong. His bound is dependent on the first maximum of a certain sequence of radicals defined using the absolute value of the coefficients of the polynomial. In the univariate setting, a bound due to Lagrange considers the first and the second maximum in the same radical sequence and is shown by Collins to be better than Hong's bound. In the 1930's, Westerfield had proposed a bound that consides every value in the same radical sequence and improves on Lagrange's bound. In this paper, we provide a generalization of Westerfield's bound to the multivariate setting. As a specialization of this bound, we also derive a generalization of Lagrange's bound, which is a strict improvement upon Hong's bound. Finally, we give an algorithm to compute this improved bound. The running time of this algorithm matches the running time of the best known algorithm to compute Hong's bound.","PeriodicalId":204559,"journal":{"name":"Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation","volume":"308 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122735607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}