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

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Integral bases for p-recursive sequences p-递归序列的积分基
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404004
Shaoshi Chen, Lixin Du, Manuel Kauers, Thibaut Verron
{"title":"Integral bases for p-recursive sequences","authors":"Shaoshi Chen, Lixin Du, Manuel Kauers, Thibaut Verron","doi":"10.1145/3373207.3404004","DOIUrl":"https://doi.org/10.1145/3373207.3404004","url":null,"abstract":"In an earlier paper, the notion of integrality known for algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes not only the algebraic and the D-finite cases but also covers the case of P-recursive sequences.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"214 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122380393","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
Parametric standard system for mixed module and its application to singularity theory 混合模组参数标准体系及其在奇点理论中的应用
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404027
H. Teramoto, Katsusuke Nabeshima
{"title":"Parametric standard system for mixed module and its application to singularity theory","authors":"H. Teramoto, Katsusuke Nabeshima","doi":"10.1145/3373207.3404027","DOIUrl":"https://doi.org/10.1145/3373207.3404027","url":null,"abstract":"We provide a concrete computational algorithm for computing the standard basis for a mixed module proposed by Gatermann and Hosten [1]. We extend it to parametric standard system for a mixed module and provide an algorithm to compute it. We demonstrate our algorithm by applying it to classification of map-germs relative to A in which complicated moduli structures appear.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124025613","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
On the bit complexity of finding points in connected components of a smooth real hypersurface 光滑实超曲面连通分量中寻点的位复杂度
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404058
J. Elliott, M. Giesbrecht, É. Schost
{"title":"On the bit complexity of finding points in connected components of a smooth real hypersurface","authors":"J. Elliott, M. Giesbrecht, É. Schost","doi":"10.1145/3373207.3404058","DOIUrl":"https://doi.org/10.1145/3373207.3404058","url":null,"abstract":"We present a full analysis of the bit complexity of an efficient algorithm for the computation of at least one point in each connected component of a smooth real hypersurface. This is a basic and important operation in semi-algebraic geometry: it gives an upper bound on the number of connected components of a real hypersurface, and is also used in many higher level algorithms. Our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proved to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; the analysis of the bit complexity and the error probability were left for future work. Our paper answers these questions. Our main contribution is a quantitative analysis of several genericity statements, such as Thom's weak transversality theorem or Noether normalization properties for polar varieties.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132756106","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
On the chordality of ordinary differential triangular decomposition in top-down style 自顶向下常微分三角形分解的弦性
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3403999
Chenqi Mou
{"title":"On the chordality of ordinary differential triangular decomposition in top-down style","authors":"Chenqi Mou","doi":"10.1145/3373207.3403999","DOIUrl":"https://doi.org/10.1145/3373207.3403999","url":null,"abstract":"In this paper we extend existing theoretical results on chordal graphs in algebraic triangular decomposition in top-down style to the ordinary differential case. We first propose the concept of differential associated graph of an ordinary differential polynomial set, and then for two typical algorithms in top-down style for ordinary differential triangular decomposition based on the pseudo-division and subresultant regular subchain respectively, we prove that when the input differential polynomial set has a chordal differential associated graph G and one perfect elimination ordering of G is used, the differential associated graph of any polynomial set in the decomposition process by these two algorithms is a subgraph of G.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129019677","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: 0
New progress in univariate polynomial root finding 一元多项式求根的新进展
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404063
Rémi Imbach, V. Pan
{"title":"New progress in univariate polynomial root finding","authors":"Rémi Imbach, V. Pan","doi":"10.1145/3373207.3404063","DOIUrl":"https://doi.org/10.1145/3373207.3404063","url":null,"abstract":"The recent advanced sub-division algorithm is nearly optimal for the approximation of the roots of a dense polynomial given in monomial basis; moreover, it works locally and slightly outperforms the user's choice MPSolve when the initial region of interest contains a small number of roots. Its basic and bottleneck block is counting the roots in a given disc on the complex plain based on Pellet's theorem, which requires the coefficients of the polynomial and expensive shift of the variable. We implement a novel method for both root-counting and exclusion test, which is faster, avoids the above requirements, and remains efficient for sparse input polynomials. It relies on approximation of the power sums of the roots lying in the disc rather than on Pellet's theorem. Such approximation was used by Schönhage in 1982 for the different task of deflation of a factor of a polynomial provided that the boundary circle of the disc is sufficiently well isolated from the roots. We implement a faster version of root-counting and exclusion test where we do not verify isolation and significantly improve performance of subdivision algorithms, particularly strongly in the case of sparse inputs. We present our implementation as heuristic and cite some relevant results on its formal support presented elsewhere.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128149080","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: 5
The orbiter ecosystem for combinatorial data 组合数据的轨道生态系统
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3403984
Anton Betten
{"title":"The orbiter ecosystem for combinatorial data","authors":"Anton Betten","doi":"10.1145/3373207.3403984","DOIUrl":"https://doi.org/10.1145/3373207.3403984","url":null,"abstract":"We describe a very versatile, fast and useful open source software package to compute combinatorial objects up to isomorphism called Orbiter. We provide an overview of some of the design decisions made during development, and we point out similar software packages. We discuss ways in which combinatorial data can be computed, analyzed and permanently stored for later use. This paper expands on earlier work published in [8].","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124957836","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
Further results on the factorization and equivalence for multivariate polynomial matrices 多元多项式矩阵的分解与等价的进一步结果
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404020
Dong Lu, Dingkang Wang, Fanghui Xiao
{"title":"Further results on the factorization and equivalence for multivariate polynomial matrices","authors":"Dong Lu, Dingkang Wang, Fanghui Xiao","doi":"10.1145/3373207.3404020","DOIUrl":"https://doi.org/10.1145/3373207.3404020","url":null,"abstract":"This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present a new criterion for the existence of matrix factorizations for a class of multivariate polynomial matrices, and prove that these matrix factorizations are unique. Based on this new criterion and the constructive proof process, we give an algorithm to compute a matrix factorization of a multivariate polynomial matrix. After that, we put forward a sufficient and necessary condition for the equivalence of square polynomial matrices: a square polynomial matrix is equivalent to a diagonal triangle if it satisfies the condition. An illustrative example is given to show the effectiveness of the matrix equivalence theorem.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133226722","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
Sparse multiplication for skew polynomials 斜多项式的稀疏乘法
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404023
M. Giesbrecht, Qiao-Long Huang, É. Schost
{"title":"Sparse multiplication for skew polynomials","authors":"M. Giesbrecht, Qiao-Long Huang, É. Schost","doi":"10.1145/3373207.3404023","DOIUrl":"https://doi.org/10.1145/3373207.3404023","url":null,"abstract":"Consider the skew polynomial ring L[x; σ], where L is a field and σ is an automorphism of L of order r. We present two randomized algorithms for the multiplication of sparse skew polynomials in L[x;σ]. The first algorithm is Las Vegas; it relies on evaluation and interpolation on a normal basis, at successive powers of a normal element. For inputs A, B ∈ L[x; σ] of degrees at most d, its expected runtime is O~(max(d,r)rRω-2) operations in K, where K = Lσ is the fixed field of σ in L and R ≤ r is the size of the Minkowski sum supp(A) + supp(B) taken modulo r; here, the supports supp(A), supp(B) are the exponents of non-zero terms in A and B. The second algorithm is Monte Carlo; it is \"super-sparse\", in the sense that its expected runtime is O~(log(d)Srω), where S is the size of supp(A) + supp(B). Using a suitable form of Kronecker substitution, we extend this second algorithm to handle multivariate polynomials, for certain families of extensions.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116044733","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
Some properties of multivariate differential dimension polynomials and their invariants 多元微分维多项式的一些性质及其不变量
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404013
A. Levin
{"title":"Some properties of multivariate differential dimension polynomials and their invariants","authors":"A. Levin","doi":"10.1145/3373207.3404013","DOIUrl":"https://doi.org/10.1145/3373207.3404013","url":null,"abstract":"In this paper we obtain new results on multivariate dimension polynomials of differential field extensions associated with partitions of basic sets of derivations. We prove that the coefficient of the summand of the highest possible degree in the canonical representation of such a polynomial is equal to the differential transcendence degree of the extension. We also give necessary and sufficient conditions under which the multivariate dimension polynomial of a differential field extension of a given differential transcendence degree has the simplest possible form. Furthermore, we describe some relationships between a multivariate dimension polynomial of a differential field extension and dimensional characteristics of subextensions defined by subsets of the basic sets of derivations.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130553086","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: 0
Geometric modeling and regularization of algebraic problems 代数问题的几何建模和正则化
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation Pub Date : 2020-07-20 DOI: 10.1145/3373207.3404066
Zhonggang Zeng
{"title":"Geometric modeling and regularization of algebraic problems","authors":"Zhonggang Zeng","doi":"10.1145/3373207.3404066","DOIUrl":"https://doi.org/10.1145/3373207.3404066","url":null,"abstract":"Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary equations, the models can be formulated to satisfy four easily verifiable conditions so that the data form complex analytic manifolds on which the solutions maintain their structures and the Lipschitz continuity. When such a problem is given with empirical data, solving the system becomes a least squares problem whose solution uniquely exists and enjoys Lipschitz continuity as long as the data point is in a tubular neighborhood of the manifold. As a result, the singular problem is regularized as a well-posed computational problem.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131320376","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
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