Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation最新文献_第6页

Punctual Hilbert scheme and certified approximate singularities 准时希尔伯特格式与证明近似奇点

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-14 DOI: 10.1145/3373207.3404024

Angelos Mantzaflaris, B. Mourrain, Á. Szántó

{"title":"Punctual Hilbert scheme and certified approximate singularities","authors":"Angelos Mantzaflaris, B. Mourrain, Á. Szántó","doi":"10.1145/3373207.3404024","DOIUrl":"https://doi.org/10.1145/3373207.3404024","url":null,"abstract":"In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f1, ..., fN) ∈ C[x1, ..., xn]N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123690899","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}

引用次数: 4

Sub-quadratic time for riemann-roch spaces: case of smooth divisors over nodal plane projective curves riemann-roch空间的次二次时间:节点平面投影曲线上光滑因子的情况

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-13 DOI: 10.1145/3373207.3404053

S. Abelard, Alain Couvreur, Grégoire Lecerf

引用次数: 5

On the apolar algebra of a product of linear forms 关于线性形式乘积的极代数

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-12 DOI: 10.1145/3373207.3404014

Michael DiPasquale, Zachary Flores, C. Peterson

{"title":"On the apolar algebra of a product of linear forms","authors":"Michael DiPasquale, Zachary Flores, C. Peterson","doi":"10.1145/3373207.3404014","DOIUrl":"https://doi.org/10.1145/3373207.3404014","url":null,"abstract":"Apolarity is a important tool in commutative algebra and algebraic geometry which studies a form, f, by the action of polynomial differential operators on f. The quotient of all polynomial differential operators by those which annihilate f is called the apolar algebra of f. In general, the apolar algebra of a form is useful for determining its Waring decomposition, which consists of writing the form as a sum of powers of linear forms with as few summands as possible. In this article we study the apolar algebra of a product of linear forms, which generalizes the case of monomials and connects to the geometry of hyperplane arrangements. In the first part of the article we provide a bound on the Waring rank of a product of linear forms under certain genericity assumptions; for this we use the defining equations of so-called star configurations due to Geramita, Harbourne, and Migliore. In the second part of the article we use the computer algebra system Bertini, which operates by homotopy continuation methods, to solve certain rank equations for catalecticant matrices. Our computations suggest that, up to a change of variables, there are exactly six homogeneous polynomials of degree six in three variables which factor completely as a product of linear forms defining an irreducible multi-arrangement and whose apolar algebras have dimension six in degree three. As a consequence of these calculations, we find six cases of such forms with cactus rank six, five of which also have Waring rank six. Among these are products defining subarrangements of the braid and Hessian arrangements.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129508659","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}

引用次数: 3

Signature-based algorithms for Gröbner bases over tate algebras 状态代数上Gröbner基的基于签名的算法

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-10 DOI: 10.1145/3373207.3404035

X. Caruso, Tristan Vaccon, Thibaut Verron

引用次数: 5

Compatible rewriting of noncommutative polynomials for proving operator identities 证明算子恒等式的非交换多项式相容重写

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-10 DOI: 10.1145/3373207.3404047

Cyrille Chenavier, Clemens Hofstadler, C. Raab, G. Regensburger

引用次数: 7

Efficient ECM factorization in parallel with the lyness map 有效的ECM分解与lyness映射并行

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-05 DOI: 10.1145/3373207.3404044

A. Hone

{"title":"Efficient ECM factorization in parallel with the lyness map","authors":"A. Hone","doi":"10.1145/3373207.3404044","DOIUrl":"https://doi.org/10.1145/3373207.3404044","url":null,"abstract":"The Lyness map is a birational map in the plane which provides one of the simplest discrete analogues of a Hamiltonian system with one degree of freedom, having a conserved quantity and an invariant symplectic form. As an example of a symmetric Quispel-Roberts-Thompson (QRT) map, each generic orbit of the Lyness map lies on a curve of genus one, and corresponds to a sequence of points on an elliptic curve which is one of the fibres in a pencil of biquadratic curves in the plane. Here we present a version of the elliptic curve method (ECM) for integer factorization, which is based on iteration of the Lyness map with a particular choice of initial data. More precisely, we give an algorithm for scalar multiplication of a point on an arbitrary elliptic curve over Q, which is represented by one of the curves in the Lyness pencil. In order to avoid field inversion (I), and require only field multiplication (M), squaring (S) and addition, projective coordinates in P1 ×P1 are used. Neglecting multiplication by curve constants (assumed small), each addition of the chosen point uses 2M, while each doubling step requires 15M. We further show that the doubling step can be implemented efficiently in parallel with four processors, dropping the effective cost to 4M. In contrast, the fastest algorithms in the literature use twisted Edwards curves (equivalent to Montgomery curves), which correspond to a subset of all elliptic curves. Scalar muliplication on twisted Edwards curves with suitable small curve constants uses 8M for point addition and 4M+4S for point doubling, both of which can be run in parallel with four processors to yield effective costs of 2M and 1M + 1S, respectively. Thus our scalar multiplication algorithm should require, on average, roughly twice as many multiplications per bit as state of the art methods using twisted Edwards curves. In our conclusions, we discuss applications where the use of Lyness curves may provide potential advantages.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130479086","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}

引用次数: 2

Separating variables in bivariate polynomial ideals 二元多项式理想中分离变量

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-04 DOI: 10.1145/3373207.3404028

Manfred Buchacher, Manuel Kauers, G. Pogudin

引用次数: 1

General witness sets for numerical algebraic geometry 数值代数几何的一般见证集

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-01 DOI: 10.1145/3373207.3403995

F. Sottile

引用次数: 3

Essentially optimal sparse polynomial multiplication 本质上是最优的稀疏多项式乘法

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-01-31 DOI: 10.1145/3373207.3404026

Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray

引用次数: 10

On fast multiplication of a matrix by its transpose 关于矩阵的转置的快速乘法

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-01-13 DOI: 10.1145/3373207.3404021

J. Dumas, Clément Pernet, A. Sedoglavic

引用次数: 5