The proof-of-work interactive protocol by Shafi Goldwasser, Yael T. Kalai and Guy N. Rothblum (GKR) [STOC 2008, JACM 2015] certifies the execution of an algorithm via the evaluation of a corresponding boolean or arithmetic circuit whose structure is known to the verifier by circuit wiring algorithms that define the uniformity of the circuit. Here we study protocols whose prover time- and space-complexities are within a poly-logarithmic factor of the time- and space-complexity of the algorithm; we call those protocols 'prover-nearly-optimal.' We show that the uniformity assumptions can be relaxed from LOGSPACE to polynomial-time in the bit-lengths of the labels which enumerate the nodes in the circuit. Our protocol applies GKR recursively to the arising sumcheck problems on each level of the circuit whose values are verified, and deploys any of the prover-nearly-optimal versions of GKR on the constructed sorting/prefix circuits with log-depth wiring functions. The verifier time-complexity of GKR grows linearly in the depth of the circuit. For deep circuits such as the Miller-Rabin integer primality test of an n-bit integer, the large number of rounds may interfere with soundness guarantees after the application of the Fiat-Shamir heuristic. We re-arrange the circuit evaluation problem by the baby-steps/giant-steps method to achieve a depth of n1/2+o(1), at prover cost n2+o(1) bit complexity and communication and verifier cost n3/2+o(1).
Shafi Goldwasser, Yael T. Kalai和Guy N. Rothblum (GKR)的工作量证明交互协议[STOC 2008, JACM 2015]通过对相应布尔或算术电路的评估来证明算法的执行，该电路的结构通过定义电路均匀性的电路布线算法为验证者所知。在这里，我们研究了证明者的时间和空间复杂性在算法的时间和空间复杂性的多对数因子内的协议;我们称这些协议为“已被证明接近最优的”。我们证明了均匀性假设可以在枚举电路中节点的标签的位长度上从LOGSPACE放宽到多项式时间。我们的协议将GKR递归地应用于每一层电路上的sumcheck问题，这些问题的值被验证，并在构造的排序/前缀电路上部署任何证明的近最优版本的GKR，这些电路具有对数深度的布线功能。GKR的验证器时间复杂度随电路深度呈线性增长。对于深度电路，如n位整数的Miller-Rabin整数素数检验，在应用Fiat-Shamir启发式后，大量的轮数可能会干扰可靠性保证。我们用小步/大步法重新安排电路评估问题，使其深度达到n1/2+o(1)，证明者代价为n2+o(1)位复杂度，通信和验证者代价为n3/2+o(1)。
{"title":"The GKR Protocol Revisited: Nearly Optimal Prover-Complexity for Polynomial-Time Wiring Algorithms and for Primality Testing in n1/2+o(1) Rounds","authors":"E. Kaltofen","doi":"10.1145/3476446.3536183","DOIUrl":"https://doi.org/10.1145/3476446.3536183","url":null,"abstract":"The proof-of-work interactive protocol by Shafi Goldwasser, Yael T. Kalai and Guy N. Rothblum (GKR) [STOC 2008, JACM 2015] certifies the execution of an algorithm via the evaluation of a corresponding boolean or arithmetic circuit whose structure is known to the verifier by circuit wiring algorithms that define the uniformity of the circuit. Here we study protocols whose prover time- and space-complexities are within a poly-logarithmic factor of the time- and space-complexity of the algorithm; we call those protocols 'prover-nearly-optimal.' We show that the uniformity assumptions can be relaxed from LOGSPACE to polynomial-time in the bit-lengths of the labels which enumerate the nodes in the circuit. Our protocol applies GKR recursively to the arising sumcheck problems on each level of the circuit whose values are verified, and deploys any of the prover-nearly-optimal versions of GKR on the constructed sorting/prefix circuits with log-depth wiring functions. The verifier time-complexity of GKR grows linearly in the depth of the circuit. For deep circuits such as the Miller-Rabin integer primality test of an n-bit integer, the large number of rounds may interfere with soundness guarantees after the application of the Fiat-Shamir heuristic. We re-arrange the circuit evaluation problem by the baby-steps/giant-steps method to achieve a depth of n1/2+o(1), at prover cost n2+o(1) bit complexity and communication and verifier cost n3/2+o(1).","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130822686","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}
A sequence is called C-finite, if it satisfies a linear recurrence with constant coefficients and holonomic, if it satisfies a linear recurrence with polynomial coefficients. The class of C2-finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients whose leading coefficient has no zero terms. Recently, we investigated computational properties of $C^2$-finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring. From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple C2-finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function.
{"title":"Simple C2-finite Sequences: a Computable Generalization of C-finite Sequences","authors":"P. Nuspl, V. Pillwein","doi":"10.1145/3476446.3536174","DOIUrl":"https://doi.org/10.1145/3476446.3536174","url":null,"abstract":"A sequence is called C-finite, if it satisfies a linear recurrence with constant coefficients and holonomic, if it satisfies a linear recurrence with polynomial coefficients. The class of C2-finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients whose leading coefficient has no zero terms. Recently, we investigated computational properties of $C^2$-finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring. From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple C2-finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121178501","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}
Evangelos Bartzos, I. Emiris, I. Kotsireas, C. Tzamos
Determining the number of solutions of a multi-homogeneous polynomial system is a fundamental problem in algebraic geometry. The multi-homogeneous Bézout (m-Bézout) number bounds from above the number of non-singular solutions of a multi-homogeneous system, but its computation is a #P>-hard problem. Recent work related the m-Bézout number of certain multi-homogeneous systems derived from rigidity theory with graph orientations, cf Bartzos et al. (2020). A first generalization applied graph orientations for bounding the root count of a multi-homogeneous system that can be modeled by simple undirected graphs, as shown by three of the authors (Bartzos et al., 2021). Here, we prove that every multi-homogeneous system can be modeled by hypergraphs and the computation of its m-Bézout bound is related to constrained hypergraph orientations. Thus, we convert the algebraic problem of bounding the number of roots of a polynomial system to a purely combinatorial problem of analyzing the structure of a hypergraph. We also provide a formulation of the orientation problem as a constraint satisfaction problem (CSP), hence leading to an algorithm that computes the multi-homogeneous bound by finding constrained hypergraph orientations.
确定多齐次多项式系统解的个数是代数几何中的一个基本问题。多齐次bsamzout (m- bsamzout)数界是从多齐次系统的非奇异解的个数出发的，但其计算是一个#P>-困难的问题。最近的工作涉及从具有图取向的刚性理论推导出的某些多齐次系统的m- bsamzout数，参见Bartzos et al.(2020)。第一个推广应用图方向来限定多齐次系统的根计数，该系统可以通过简单的无向图建模，如三位作者所示(Bartzos et al.， 2021)。本文证明了每一个多齐次系统都可以用超图来建模，并且它的m- bsamzout界的计算与约束超图的方向有关。因此，我们将多项式系统的根数限定的代数问题转化为分析超图结构的纯组合问题。我们还提供了方向问题作为约束满足问题(CSP)的公式，从而导致通过寻找约束超图方向来计算多齐次界的算法。
{"title":"Bounding the Number of Roots of Multi-Homogeneous Systems","authors":"Evangelos Bartzos, I. Emiris, I. Kotsireas, C. Tzamos","doi":"10.1145/3476446.3536189","DOIUrl":"https://doi.org/10.1145/3476446.3536189","url":null,"abstract":"Determining the number of solutions of a multi-homogeneous polynomial system is a fundamental problem in algebraic geometry. The multi-homogeneous Bézout (m-Bézout) number bounds from above the number of non-singular solutions of a multi-homogeneous system, but its computation is a #P>-hard problem. Recent work related the m-Bézout number of certain multi-homogeneous systems derived from rigidity theory with graph orientations, cf Bartzos et al. (2020). A first generalization applied graph orientations for bounding the root count of a multi-homogeneous system that can be modeled by simple undirected graphs, as shown by three of the authors (Bartzos et al., 2021). Here, we prove that every multi-homogeneous system can be modeled by hypergraphs and the computation of its m-Bézout bound is related to constrained hypergraph orientations. Thus, we convert the algebraic problem of bounding the number of roots of a polynomial system to a purely combinatorial problem of analyzing the structure of a hypergraph. We also provide a formulation of the orientation problem as a constraint satisfaction problem (CSP), hence leading to an algorithm that computes the multi-homogeneous bound by finding constrained hypergraph orientations.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125508951","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}
Consider an integer matrix A ε Znx(n-1) that has full column rank n-1. The set of all Z-linear combinations of the rows of A generates a lattice, denoted by L(A).
{"title":"Computing a Basis for an Integer Lattice: A Special Case","authors":"Haomin Li, A. Storjohann","doi":"10.1145/3476446.3536184","DOIUrl":"https://doi.org/10.1145/3476446.3536184","url":null,"abstract":"Consider an integer matrix A ε Znx(n-1) that has full column rank n-1. The set of all Z-linear combinations of the rows of A generates a lattice, denoted by L(A).","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116261691","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}
In Commutative Algebra and Algebraic Geometry, ''Primary decomposition'' is well-known as a fundamental and important tool. Although algorithms for primary decomposition have been studied by many researchers, the development of fast algorithms still remains a challenging problem. In this paper, we devise an algorithm for ''Strong Intermediate Primary Decomposition" via maximal independent sets by using modular techniques. In the algorithm, we utilize double ideal quotients to check whether a candidate from modular computations is an intersection of prime divisors or not. As an application, we can compute the set of associated prime divisors from the strong intermediate prime decomposition. In a naive computational experiment, we see the effectiveness of our methods.
{"title":"Modular Techniques for Intermediate Primary Decomposition","authors":"Yuki Ishihara","doi":"10.1145/3476446.3535488","DOIUrl":"https://doi.org/10.1145/3476446.3535488","url":null,"abstract":"In Commutative Algebra and Algebraic Geometry, ''Primary decomposition'' is well-known as a fundamental and important tool. Although algorithms for primary decomposition have been studied by many researchers, the development of fast algorithms still remains a challenging problem. In this paper, we devise an algorithm for ''Strong Intermediate Primary Decomposition\" via maximal independent sets by using modular techniques. In the algorithm, we utilize double ideal quotients to check whether a candidate from modular computations is an intersection of prime divisors or not. As an application, we can compute the set of associated prime divisors from the strong intermediate prime decomposition. In a naive computational experiment, we see the effectiveness of our methods.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121849904","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}
We briefly describe a flurry of recent activity in the interaction between the theory of computation and several mathematical areas, that has led to many applications on both sides. The core results are mainly new algorithms for basic problems in invariant theory, arising from computational questions in algebraic complexity theory. However, as understanding evolved, connections were revealed to many other mathematical disciplines, as well as to optimization theory. In particular, the most basic tools of convex optimization in Euclidean space extend to a far more general geodesic setting of Riemannian manifolds that arise from the symmetries of non-commutative groups. This paper extends a section in my book, Mathematics and Computation [54] devoted to an accessible exposition of the theory of computation. Besides covering many of the different parts of this theory, the book discusses its connections with many different areas of mathematics, and many of the sciences.
我们简要地描述了最近计算理论和几个数学领域之间相互作用的一系列活动，这导致了双方的许多应用。核心成果主要是由代数复杂性理论中的计算问题引起的不变量理论基本问题的新算法。然而，随着理解的发展，联系被揭示给许多其他数学学科，以及优化理论。特别是，欧几里得空间中凸优化的最基本工具扩展到由非交换群的对称性产生的黎曼流形的更一般的测地线集合。本文扩展了我的书《数学与计算》(Mathematics and Computation[54])中的一个章节，专门用于对计算理论进行通俗易懂的阐述。除了涵盖这一理论的许多不同部分，书中讨论了它的连接与许多不同的数学领域，和许多科学。
{"title":"Non-commutative Optimization - Where Algebra, Analysis and Computational Complexity Meet","authors":"A. Wigderson","doi":"10.1145/3476446.3535489","DOIUrl":"https://doi.org/10.1145/3476446.3535489","url":null,"abstract":"We briefly describe a flurry of recent activity in the interaction between the theory of computation and several mathematical areas, that has led to many applications on both sides. The core results are mainly new algorithms for basic problems in invariant theory, arising from computational questions in algebraic complexity theory. However, as understanding evolved, connections were revealed to many other mathematical disciplines, as well as to optimization theory. In particular, the most basic tools of convex optimization in Euclidean space extend to a far more general geodesic setting of Riemannian manifolds that arise from the symmetries of non-commutative groups. This paper extends a section in my book, Mathematics and Computation [54] devoted to an accessible exposition of the theory of computation. Besides covering many of the different parts of this theory, the book discusses its connections with many different areas of mathematics, and many of the sciences.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126122438","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}
This paper is concerned with a property of modules over a polynomial ring and its application in multivariate polynomial matrix factorizations. We construct a specific polynomial such that the product of the polynomial and a nonzero vector in a module over a polynomial ring can be represented by the elements in a maximum linearly independent vector set of the module over the polynomial ring. Based on this property, a relationship between a rank-deficient matrix and any of its full row rank submatrices is presented. By this result, we show that the problem for general factorizations of rank-deficient matrices can be translated into that of any of their full row rank submatrices in the regular case. Then many results on factorizations of full row rank matrices, such as zero prime factorizations, minor prime factorizations and factor prime factorizations, can be extended to the rank-deficient case. We implement the algorithm of general factorizations for rank-deficient matrices on the computer algebra system Maple, and two examples are given to illustrate the algorithm.
{"title":"A Property of Modules Over a Polynomial Ring With an Application in Multivariate Polynomial Matrix Factorizations","authors":"Dong Lu, Dingkang Wang, Fanghui Xiao, Xiaopeng Zheng","doi":"10.1145/3476446.3535470","DOIUrl":"https://doi.org/10.1145/3476446.3535470","url":null,"abstract":"This paper is concerned with a property of modules over a polynomial ring and its application in multivariate polynomial matrix factorizations. We construct a specific polynomial such that the product of the polynomial and a nonzero vector in a module over a polynomial ring can be represented by the elements in a maximum linearly independent vector set of the module over the polynomial ring. Based on this property, a relationship between a rank-deficient matrix and any of its full row rank submatrices is presented. By this result, we show that the problem for general factorizations of rank-deficient matrices can be translated into that of any of their full row rank submatrices in the regular case. Then many results on factorizations of full row rank matrices, such as zero prime factorizations, minor prime factorizations and factor prime factorizations, can be extended to the rank-deficient case. We implement the algorithm of general factorizations for rank-deficient matrices on the computer algebra system Maple, and two examples are given to illustrate the algorithm.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133462605","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}
Polynomial matrices and ideals generated by their minors appear in various domains such as cryptography, polynomial optimization and effective algebraic geometry. When the given matrix is symmetric, this additional structure on top of the determinantal structure, affects computations on the derived ideals. Thus, understanding the complexity of these computations is important. Moreover, this study serves as a stepping stone towards further understanding the effects of structure in determinantal systems, such as those coming from moment matrices. In this paper, we focus on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals. Under a variant of Fröberg's conjecture, we study its complexity for symmetric determinantal ideals and identify the gain of exploiting sparsity in the Sparse-FGLM algorithm compared with the classical FGLM algorithm. For an n×n symmetric matrix with polynomial entries of degree d, we show that the complexity of Sparse-FGLM for zero-dimensional determinantal ideals obtained from this matrix over that of the FGLM algorithm is at least O(1/d). Moreover, for some specific sizes of minors, we prove finer results of at least O(1/nd) and O(1/m3d).
{"title":"Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals","authors":"A. Ferguson, H. P. Le","doi":"10.1145/3476446.3536182","DOIUrl":"https://doi.org/10.1145/3476446.3536182","url":null,"abstract":"Polynomial matrices and ideals generated by their minors appear in various domains such as cryptography, polynomial optimization and effective algebraic geometry. When the given matrix is symmetric, this additional structure on top of the determinantal structure, affects computations on the derived ideals. Thus, understanding the complexity of these computations is important. Moreover, this study serves as a stepping stone towards further understanding the effects of structure in determinantal systems, such as those coming from moment matrices. In this paper, we focus on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals. Under a variant of Fröberg's conjecture, we study its complexity for symmetric determinantal ideals and identify the gain of exploiting sparsity in the Sparse-FGLM algorithm compared with the classical FGLM algorithm. For an n×n symmetric matrix with polynomial entries of degree d, we show that the complexity of Sparse-FGLM for zero-dimensional determinantal ideals obtained from this matrix over that of the FGLM algorithm is at least O(1/d). Moreover, for some specific sizes of minors, we prove finer results of at least O(1/nd) and O(1/m3d).","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":" 36","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114060703","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}
Given a linear equation E, the k-color Rado number Rk(E) is the smallest integer n such that every k-coloring of {1,2,3,...,n} contains a monochromatic solution to E. The degree of regularity of E, denoted dor(E), is the largest value k such that Rk(E) is finite. In this article we present new theoretical and computational results about the Rado numbers R3(E) and the degree of regularity of three-variable equations E.
{"title":"Rado Numbers and SAT Computations","authors":"Yuan Chang, J. D. Loera, W. J. Wesley","doi":"10.1145/3476446.3535494","DOIUrl":"https://doi.org/10.1145/3476446.3535494","url":null,"abstract":"Given a linear equation E, the k-color Rado number Rk(E) is the smallest integer n such that every k-coloring of {1,2,3,...,n} contains a monochromatic solution to E. The degree of regularity of E, denoted dor(E), is the largest value k such that Rk(E) is finite. In this article we present new theoretical and computational results about the Rado numbers R3(E) and the degree of regularity of three-variable equations E.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1088 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123339305","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}
An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently based on the extended subresultant theorem for parametric cases, the separating element corresponding to each branch is chosen with the further partition of parameter space. Finally, with the help of parametric greatest common divisor theory a finite set of the rational univariate representation of zero-dimensional ideals with parameters can be obtained.
{"title":"Rational Univariate Representation of Zero-Dimensional Ideals with Parameters","authors":"Dingkang Wang, Jingjing Wei, Fanghui Xiao, Xiaopeng Zheng","doi":"10.1145/3476446.3535496","DOIUrl":"https://doi.org/10.1145/3476446.3535496","url":null,"abstract":"An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently based on the extended subresultant theorem for parametric cases, the separating element corresponding to each branch is chosen with the further partition of parameter space. Finally, with the help of parametric greatest common divisor theory a finite set of the rational univariate representation of zero-dimensional ideals with parameters can be obtained.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130565716","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}
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