Let P ∈ Z[X, Y] be a square-free polynomial and C(P):= {(α, β) ∈ R2, P(α, β) = 0} be the real algebraic curve defined by P. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular points and critical points of the projection wrt the X-axis in Õ(d6τ+d7) bit operations where Õ means that we ignore logarithmic factors in d and τ. Compared to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of C(P) i.e a straight-line planar graph isotopic to C(P) in Õ(d6τ + d7) bit operations.
{"title":"On the computation of the topology of plane curves","authors":"D. Diatta, F. Rouillier, Marie-Françoise Roy","doi":"10.1145/2608628.2608670","DOIUrl":"https://doi.org/10.1145/2608628.2608670","url":null,"abstract":"Let <i>P</i> ∈ Z[<i>X, Y</i>] be a square-free polynomial and C(<i>P</i>):= {(α, β) ∈ R<sup>2</sup>, <i>P</i>(α, β) = 0} be the real algebraic curve defined by <i>P</i>. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular points and critical points of the projection wrt the <i>X</i>-axis in <i>Õ</i>(<i>d</i><sup>6</sup>τ+<i>d</i><sup>7</sup>) bit operations where <i>Õ</i> means that we ignore logarithmic factors in <i>d</i> and <i>τ</i>. Compared to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of C(<i>P</i>) <i>i.e</i> a straight-line planar graph isotopic to C(<i>P</i>) in <i>Õ</i>(<i>d</i><sup>6</sup><i>τ</i> + <i>d</i><sup>7</sup>) bit operations.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"237 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125065380","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 tutorial will focus on the basics of max-plus algebra along with relevant topics. Max-plus algebra is a discrete algebraic system in which the max and plus operations are defined as addition and multiplication in conventional algebra. Using this system, the behavior of a class of discrete event systems can be represented by simple linear equations, by which modeling, analysis, and control of the systems can be realized. We will start with a simple project scheduling problem to understand the basic usage of max-plus algebra. The focus will then be oriented to its detailed definition and observe relevant properties in terms of graph theory, net theory, and so on. In the latter part, we will move on to modeling and formulation methods in control theory viewpoint. Several application examples as schedule solvers will be introduced, followed by several recent advances achieved by the presenter.
{"title":"Introduction to max-plus algebra","authors":"H. Goto","doi":"10.1145/2608628.2627496","DOIUrl":"https://doi.org/10.1145/2608628.2627496","url":null,"abstract":"This tutorial will focus on the basics of max-plus algebra along with relevant topics. Max-plus algebra is a discrete algebraic system in which the max and plus operations are defined as addition and multiplication in conventional algebra. Using this system, the behavior of a class of discrete event systems can be represented by simple linear equations, by which modeling, analysis, and control of the systems can be realized.\u0000 We will start with a simple project scheduling problem to understand the basic usage of max-plus algebra. The focus will then be oriented to its detailed definition and observe relevant properties in terms of graph theory, net theory, and so on. In the latter part, we will move on to modeling and formulation methods in control theory viewpoint. Several application examples as schedule solvers will be introduced, followed by several recent advances achieved by the presenter.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122166616","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}
When David Hilbert started so called "Hilbert's program" (formalization of mathematics) in the early 20th century to give a solid foundation to mathematics, he unintentionally introduced the possibility of automatization of mathematics. Theoretically, the possibility was denied by Gödel's incompleteness theorem. However, an interesting issue remains: is "mundane mathematics" automatizable? We are developing a system that solves a wide range of math problems written in natural language, as a part of the Todai Robot Project, an AI challenge to pass the university entrance examination. We give an overview and report on the progress of our project, and the theoretical and methodological difficulties to be overcome.
{"title":"Mathematics by machine","authors":"N. Arai, Takuya Matsuzaki, Hidenao Iwane, H. Anai","doi":"10.1145/2608628.2627488","DOIUrl":"https://doi.org/10.1145/2608628.2627488","url":null,"abstract":"When David Hilbert started so called \"Hilbert's program\" (formalization of mathematics) in the early 20th century to give a solid foundation to mathematics, he unintentionally introduced the possibility of automatization of mathematics. Theoretically, the possibility was denied by Gödel's incompleteness theorem. However, an interesting issue remains: is \"mundane mathematics\" automatizable? We are developing a system that solves a wide range of math problems written in natural language, as a part of the Todai Robot Project, an AI challenge to pass the university entrance examination. We give an overview and report on the progress of our project, and the theoretical and methodological difficulties to be overcome.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"2011 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121347977","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 this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zn-graded polynomials in the nth q-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Zn-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples.
{"title":"Factoring linear differential operators in n variables","authors":"M. Giesbrecht, A. Heinle, V. Levandovskyy","doi":"10.1145/2608628.2608667","DOIUrl":"https://doi.org/10.1145/2608628.2608667","url":null,"abstract":"In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zn-graded polynomials in the nth <u>q</u>-Weyl algebra.\u0000 The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Zn-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring.\u0000 The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131522356","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 the tutorial, we will introduce two kinds of problems for which validated results are computed via hybrid symbolic-numeric algorithms. These hybrid algorithms follow the basic principle pointed out by Siegfried M. Rump in [1] for computing validated results: First, a pure floating point algorithm is used to compute an approximate solution of good quality for a given problem. Second, a verification step using exact rational arithmetic or interval arithmetic is appended. If this step is successful, then certified lower bounds or verified error bounds are computed for the previously computed approximation.
在本教程中，我们将介绍两种通过混合符号-数值算法计算验证结果的问题。这些混合算法遵循Siegfried M. Rump在[1]中指出的计算验证结果的基本原则:首先，使用纯浮点算法计算给定问题的高质量近似解。其次，使用精确有理数运算或区间运算进行验证。如果此步骤成功，则为先前计算的近似值计算经过验证的下界或经过验证的错误边界。
{"title":"Symbolic-numeric algorithms for computing validated results","authors":"L. Zhi","doi":"10.1145/2608628.2627491","DOIUrl":"https://doi.org/10.1145/2608628.2627491","url":null,"abstract":"In the tutorial, we will introduce two kinds of problems for which validated results are computed via hybrid symbolic-numeric algorithms. These hybrid algorithms follow the basic principle pointed out by Siegfried M. Rump in [1] for computing validated results: First, a pure floating point algorithm is used to compute an approximate solution of good quality for a given problem. Second, a verification step using exact rational arithmetic or interval arithmetic is appended. If this step is successful, then certified lower bounds or verified error bounds are computed for the previously computed approximation.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114791939","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 rectangular matrix F ∈ K[x]mxn with m < n of univariate polynomials over a field K. we give an efficient algorithm for computing a unimodular completion of F. Our algorithm is deterministic and computes such a completion, when it exists, with cost O~ (nωs) field operations from K. Here s is the average of the m largest column degrees of F and ω is the exponent on the cost of matrix multiplication. Here O~ is big-O but with log factors removed. If a unimodular completion does not exist for F, our algorithm computes a unimodular completion for a right cofactor of a column basis of F, or equivalently, computes a completion that preserves the generalized determinant.
{"title":"Unimodular completion of polynomial matrices","authors":"Wei Zhou, G. Labahn","doi":"10.1145/2608628.2608640","DOIUrl":"https://doi.org/10.1145/2608628.2608640","url":null,"abstract":"Given a rectangular matrix <b>F</b> ∈ K[<i>x</i>]<sup><i>m</i>x<i>n</i></sup> with <i>m</i> < <i>n</i> of univariate polynomials over a field K. we give an efficient algorithm for computing a unimodular completion of <b>F</b>. Our algorithm is deterministic and computes such a completion, when it exists, with cost <i>O</i>~ (<i>n</i><sup>ω</sup><i>s</i>) field operations from K. Here <i>s</i> is the average of the <i>m</i> largest column degrees of <b>F</b> and ω is the exponent on the cost of matrix multiplication. Here <i>O</i>~ is big-O but with log factors removed. If a unimodular completion does not exist for <b>F</b>, our algorithm computes a unimodular completion for a right cofactor of a column basis of <b>F</b>, or equivalently, computes a completion that preserves the generalized determinant.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120972140","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}
The theory of Gröbner bases has a lot of application in many research areas, and is implemented in various mathematical software; see, e.g., [2, 3]. Among their application, this tutorial will focus on basic and recent developments in the theory of Gröbner bases of toric ideals. Toric ideals have been studied for a long time. For example, in the book [9], Herzog's paper [6] was introduced as an early reference. In 1990's, several breakthroughs on toric ideals were done: • Conti--Traverso algorithm for integer programming using Gröbner bases of toric ideals (see [1]); • Correspondence between regular triangulations [5] of integral convex polytopes and Gröbner bases of toric ideals (see [8]); • Diaconis--Sturmfels algorithm for Markov chain Monte Carlo method in the examination of a statistical model using a set of generators of toric ideals (see [4]). In this tutorial, starting with introduction to Gröbner bases and toric ideals, we study some topics related with breakthroughs above. A lot of mathematical software contributed to developments of this research area. (One can find a partial list of such software in Chapters 3 and 7 of [7].)
{"title":"Gröbner bases of toric ideals and their application","authors":"Hidefumi Ohsugi","doi":"10.1145/2608628.2627495","DOIUrl":"https://doi.org/10.1145/2608628.2627495","url":null,"abstract":"The theory of Gröbner bases has a lot of application in many research areas, and is implemented in various mathematical software; see, e.g., [2, 3]. Among their application, this tutorial will focus on basic and recent developments in the theory of Gröbner bases of toric ideals. Toric ideals have been studied for a long time. For example, in the book [9], Herzog's paper [6] was introduced as an early reference. In 1990's, several breakthroughs on toric ideals were done:\u0000 • Conti--Traverso algorithm for integer programming using Gröbner bases of toric ideals (see [1]);\u0000 • Correspondence between regular triangulations [5] of integral convex polytopes and Gröbner bases of toric ideals (see [8]);\u0000 • Diaconis--Sturmfels algorithm for Markov chain Monte Carlo method in the examination of a statistical model using a set of generators of toric ideals (see [4]).\u0000 In this tutorial, starting with introduction to Gröbner bases and toric ideals, we study some topics related with breakthroughs above. A lot of mathematical software contributed to developments of this research area. (One can find a partial list of such software in Chapters 3 and 7 of [7].)","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"141 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114908360","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 extend the theory and the algorithms of Border Basis to systems of Laurent polynomial equations, defining "toric" roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals.
{"title":"Toric border basis","authors":"B. Mourrain, P. Trebuchet","doi":"10.1145/2608628.2608652","DOIUrl":"https://doi.org/10.1145/2608628.2608652","url":null,"abstract":"We extend the theory and the algorithms of Border Basis to systems of Laurent polynomial equations, defining \"toric\" roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115353143","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}
Let V ∈ Q(i)(a1,..., an)(q1, q2) be a rationally parametrized planar homogeneous potential of homogeneity degree k ≠ −2, 0, 2. We design an algorithm that computes polynomial necessary conditions on the parameters (a1,..., an) such that the dynamical system associated to the potential V is integrable. These conditions originate from those of the Morales-Ramis-Simó integrability criterion near all Darboux points. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree 9. Another striking application is the first complete proof of the non-integrability of the collinear three body problem.
{"title":"Computing necessary integrability conditions for planar parametrized homogeneous potentials","authors":"A. Bostan, Thierry Combot, M. S. E. Din","doi":"10.1145/2608628.2608662","DOIUrl":"https://doi.org/10.1145/2608628.2608662","url":null,"abstract":"Let <i>V</i> ∈ Q(<i>i</i>)(<b>a</b><sub>1</sub>,..., <b>a</b><sub><i>n</i></sub>)(<b>q</b><sub>1</sub>, <b>q</b><sub>2</sub>) be a rationally parametrized planar homogeneous potential of homogeneity degree <i>k</i> ≠ −2, 0, 2. We design an algorithm that computes polynomial <i>necessary</i> conditions on the parameters (<b>a</b><sub>1</sub>,..., <b>a</b><sub><i>n</i></sub>) such that the dynamical system associated to the potential <i>V</i> is integrable. These conditions originate from those of the Morales-Ramis-Simó integrability criterion near all Darboux points. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree 9. Another striking application is the first complete proof of the non-integrability of the <i>collinear three body problem</i>.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130914881","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}
Let (f1,..., fs) ∈ Qp [X1,..., Xn]s be a sequence of homogeneous polynomials with p-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since Ap is not an effective field, classical algorithm does not apply. We provide a definition for an approximate Gröbner basis with respect to a monomial order w. We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals ⟨f1,..., fi⟩ are weakly-w-ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic. Two variants of that strategy are available, depending on whether one lean more on precision or time-complexity. For the analysis of these algorithms, we study the loss of precision of the Gauss row-echelon algorithm, and apply it to an adapted Matrix-F5 algorithm. Numerical examples are provided.
{"title":"Matrix-F5 algorithms over finite-precision complete discrete valuation fields","authors":"Tristan Vaccon","doi":"10.1145/2608628.2608658","DOIUrl":"https://doi.org/10.1145/2608628.2608658","url":null,"abstract":"Let (<i>f</i><sub>1</sub>,..., <i>f</i><sub><i>s</i></sub>) ∈ Q<sub><i>p</i></sub> [<i>X</i><sub>1</sub>,..., <i>X</i><sub><i>n</i></sub>]<sup><i>s</i></sup> be a sequence of homogeneous polynomials with <i>p</i>-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since A<sub><i>p</i></sub> is not an effective field, classical algorithm does not apply.\u0000 We provide a definition for an approximate Gröbner basis with respect to a monomial order <i>w</i>. We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals ⟨<i>f</i><sub>1</sub>,..., <i>f</i><sub><i>i</i></sub>⟩ are weakly-<i>w</i>-ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic.\u0000 Two variants of that strategy are available, depending on whether one lean more on precision or time-complexity. For the analysis of these algorithms, we study the loss of precision of the Gauss row-echelon algorithm, and apply it to an adapted Matrix-F5 algorithm. Numerical examples are provided.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116663083","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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