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Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation最新文献

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Condition numbers for the cube. I: Univariate polynomials and hypersurfaces 立方体的条件数。1:单变量多项式与超曲面
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-06-08 DOI: 10.1145/3373207.3404054
Josué Tonelli-Cueto, Elias P. Tsigaridas
{"title":"Condition numbers for the cube. I: Univariate polynomials and hypersurfaces","authors":"Josué Tonelli-Cueto, Elias P. Tsigaridas","doi":"10.1145/3373207.3404054","DOIUrl":"https://doi.org/10.1145/3373207.3404054","url":null,"abstract":"The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube, that is a step in this direction. We present this framework for real hypersurfaces. We demonstrate its capabilities by providing a new probabilistic complexity analysis for the Plantinga-Vegter algorithm, which covers both random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. We present explicit results with structured random polynomials for problems with two or more dimensions. Additionally, we provide some estimates of the separation bound of a univariate polynomial in our current framework.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130092629","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: 9
Generalizing the davenport-mahler-mignotte bound: the weighted case 推广davenport-mahler-mignotte界:加权情况
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-05-16 DOI: 10.1145/3373207.3404016
Vikram Sharma
{"title":"Generalizing the davenport-mahler-mignotte bound: the weighted case","authors":"Vikram Sharma","doi":"10.1145/3373207.3404016","DOIUrl":"https://doi.org/10.1145/3373207.3404016","url":null,"abstract":"Root separation bounds play an important role as a complexity measure in understanding the behaviour of various algorithms in computational algebra, e.g., root isolation algorithms. A classic result in the univariate setting is the Davenport-Mahler-Mignotte (DMM) bound. One way to state the bound is to consider a directed acyclic graph (V, E) on a subset of roots of a degree d polynomial f (z) ∈ C[z], where the edges point from a root of smaller absolute value to one of larger absolute, and the in-degrees of all vertices is at most one. Then the DMM bound is an amortized lower bound on the following product: Π(α, β)∈E |α - β|. However, the lower bound involves the discriminant of the polynomial f, and becomes trivial if the polynomial is not square-free. This was resolved by Eigenwillig, 2008, by using a suitable subdiscriminant instead of the discriminant. Escorcielo-Perrucci, 2016, further dropped the in-degree constraint on the graph by using the theory of finite differences. Emiris et al., 2019, have generalized their result to handle the case where the exponent of the term |α - β| in the product is at most the multiplicity of either of the roots. In this paper, we generalize these results by allowing arbitrary positive integer weights on the edges of the graph, i.e., for a weight function w : E → Z<0, we derive an amortized lower bound on Π(α, β)∈E |α - β|w(α, β). Such a product occurs in the complexity estimates of some recent algorithms for root clustering (e.g., Becker et al., 2016), where the weights are usually some function of the multiplicity of the roots. Because of its amortized nature, our bound is arguably better than the bounds obtained by manipulating existing results to accommodate the weights.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128798348","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: 1
Numerical equality tests for rational maps and signatures of curves 有理映射和曲线特征的数值等式检验
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-05-10 DOI: 10.1145/3373207.3404050
Timothy Duff, Michael Ruddy
{"title":"Numerical equality tests for rational maps and signatures of curves","authors":"Timothy Duff, Michael Ruddy","doi":"10.1145/3373207.3404050","DOIUrl":"https://doi.org/10.1145/3373207.3404050","url":null,"abstract":"We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with \"probability-one\", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to noise.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"120 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123126502","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
Algorithmic averaging for studying periodic orbits of planar differential systems 研究平面微分系统周期轨道的算法平均
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-05-06 DOI: 10.1145/3373207.3404064
Bo Huang
{"title":"Algorithmic averaging for studying periodic orbits of planar differential systems","authors":"Bo Huang","doi":"10.1145/3373207.3404064","DOIUrl":"https://doi.org/10.1145/3373207.3404064","url":null,"abstract":"One of the main open problems in the qualitative theory of real planar differential systems is the study of limit cycles. In this article, we present an algorithmic approach for detecting how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems via the averaging method. We propose four symbolic algorithms to implement the averaging method. The first algorithm is based on the change of polar coordinates that allows one to transform a considered differential system to the normal form of averaging. The second algorithm is used to derive the solutions of certain differential systems associated to the unperturbed term of the normal of averaging. The third algorithm exploits the partial Bell polynomials and allows one to compute the integral formula of the averaged functions at any order. The last algorithm is based on the aforementioned algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The implementation of our algorithms is discussed and evaluated using several examples. The experimental results have extended the existing relevant results for certain classes of differential systems.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124064740","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: 1
Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation 通用二元多点评估、插值和预计算模组合
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-03-27 DOI: 10.1145/3373207.3404032
Vincent Neiger, J. Rosenkilde, Grigory Solomatov
{"title":"Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation","authors":"Vincent Neiger, J. Rosenkilde, Grigory Solomatov","doi":"10.1145/3373207.3404032","DOIUrl":"https://doi.org/10.1145/3373207.3404032","url":null,"abstract":"Suppose K is a large enough field and P ⊂ K2 is a fixed, generic set of points which is available for precomputation. We introduce a technique called reshaping which allows us to design quasi-linear algorithms for both: computing the evaluations of an input polynomial f ∈ K [x, y] at all points of P and computing an interpolant f ∈ K[x, y] which takes prescribed values on P and satisfies an input y-degree bound. Our genericity assumption is explicit and we prove that it holds for most point sets over a large enough field. If P violates the assumption, our algorithms still work and the performance degrades smoothly according to a distance from being generic. To show that the reshaping technique may have an impact on other related problems, we apply it to modular composition: suppose generic polynomials M ∈ K[x] and A ∈ K[x] are available for precomputation, then given an input f ∈ K[x, y] we show how to compute f (x, A(x)) rem M(x) in quasi-linear time.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133438080","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: 6
Modular techniques for effective localization and double ideal quotient 有效定位与双理想商的模技术
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-03-01 DOI: 10.1145/3373207.3404017
Yuki Ishihara
{"title":"Modular techniques for effective localization and double ideal quotient","authors":"Yuki Ishihara","doi":"10.1145/3373207.3404017","DOIUrl":"https://doi.org/10.1145/3373207.3404017","url":null,"abstract":"By double ideal quotient, we mean (I : (I : J)) where I and J are ideals. In our previous work [12], double ideal quotient and its variants are shown to be very useful for checking prime divisors and generating primary components. Combining those properties, we can compute \"direct localization\" effectively, comparing with full primary decomposition. In this paper, we apply modular techniques effectively to computation of such double ideal quotient and its variants, where first we compute them modulo several prime numbers and then lift them up over rational numbers by Chinese Remainder Theorem and rational reconstruction. As a new modular technique for double ideal quotient and its variants, we devise criteria for output from modular computations. Also, we apply modular techniques to intermediate primary decomposition. We examine the effectiveness of our modular techniques for several examples by preliminary computational experiments in Singular.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122692942","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
Fast in-place algorithms for polynomial operations: division, evaluation, interpolation 多项式运算的快速就地算法:除法，求值，插值
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-24 DOI: 10.1145/3373207.3404061
Pascal Giorgi, Bruno Grenet, Daniel S. Roche
{"title":"Fast in-place algorithms for polynomial operations: division, evaluation, interpolation","authors":"Pascal Giorgi, Bruno Grenet, Daniel S. Roche","doi":"10.1145/3373207.3404061","DOIUrl":"https://doi.org/10.1145/3373207.3404061","url":null,"abstract":"We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that such problems can be solved in (nearly) the same asymptotic time as fast polynomial multiplication. However, these reductions, even when applied to an in-place variant of fast polynomial multiplication, yield algorithms which require at least a linear amount of extra space for intermediate results. We demonstrate new in-place algorithms for the aforementioned polynomial computations which require only constant extra space and achieve the same asymptotic running time as their out-of-place counterparts. We also provide a precise complexity analysis so that all constants are made explicit, parameterized by the space usage of the underlying multiplication algorithms.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133387148","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: 8
Global optimization via the dual SONC cone and linear programming 基于双SONC锥和线性规划的全局优化
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-21 DOI: 10.1145/3373207.3404043
Mareike Dressler, Janin Heuer, Helen Naumann, T. Wolff
{"title":"Global optimization via the dual SONC cone and linear programming","authors":"Mareike Dressler, Janin Heuer, Helen Naumann, T. Wolff","doi":"10.1145/3373207.3404043","DOIUrl":"https://doi.org/10.1145/3373207.3404043","url":null,"abstract":"Using the dual cone of sums of nonnegative circuits (SONC), we provide a relaxation of the global optimization problem to minimize an exponential sum and, as a special case, a multivariate real polynomial. Our approach builds on two key observations. First, that the dual SONC cone is contained in the primal one. Hence, containment in this cone is a certificate of nonnegativity. Second, we show that membership in the dual cone can be verified by a linear program. We implement the algorithm and present initial experimental results comparing our method to existing approaches.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134581733","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: 16
On the uniqueness of simultaneous rational function reconstruction 论同时有理函数重构的唯一性
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-20 DOI: 10.1145/3373207.3404051
Eleonora Guerrini, R. Lebreton, Ilaria Zappatore
{"title":"On the uniqueness of simultaneous rational function reconstruction","authors":"Eleonora Guerrini, R. Lebreton, Ilaria Zappatore","doi":"10.1145/3373207.3404051","DOIUrl":"https://doi.org/10.1145/3373207.3404051","url":null,"abstract":"This paper focuses on the problem of reconstructing a vector of rational functions given some evaluations, or more generally given their remainders modulo different polynomials. The special case of rational functions sharing the same denominator, a.k.a. Simultaneous Rational Function Reconstruction (SRFR), has many applications from linear system solving to coding theory, provided that SRFR has a unique solution. The number of unknowns in SRFR is smaller than for a general vector of rational function. This allows one to reduce the number of evaluation points needed to guarantee the existence of a solution, possibly losing its uniqueness. In this work, we prove that uniqueness is guaranteed for a generic instance.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126033803","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
A divide-and-conquer algorithm for computing gröbner bases of syzygies in finite dimension 有限维协同结构gröbner基的分治算法
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-02-15 DOI: 10.1145/3373207.3404059
Simone Naldi, Vincent Neiger
{"title":"A divide-and-conquer algorithm for computing gröbner bases of syzygies in finite dimension","authors":"Simone Naldi, Vincent Neiger","doi":"10.1145/3373207.3404059","DOIUrl":"https://doi.org/10.1145/3373207.3404059","url":null,"abstract":"Let f1, ..., fm be elements in a quotient Rn/N which has finite dimension as a K-vector space, where R = K[X1, ..., Xr] and N is an R-submodule of Rn. We address the problem of computing a Gröbner basis of the module of syzygies of (f1, ..., fm), that is, of vectors (p1, ..., pm) ∈ Rm such that p1f1 + ... + pm fm = 0. An iterative algorithm for this problem was given by Marinari, Möller, and Mora (1993) using a dual representation of Rn/N as the kernel of a collection of linear functionals. Following this viewpoint, we design a divide-and-conquer algorithm, which can be interpreted as a generalization to several variables of Beckermann and Labahn's recursive approach for matrix Padé and rational interpolation problems. To highlight the interest of this method, we focus on the specific case of bivariate Padé approximation and show that it improves upon the best known complexity bounds.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"854 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126963869","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
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