V. Levandovskyy, Grischa Studzinski, Benjamin Schnitzler
Recently, the notion of "letterplace correspondence" between ideals in the free associative algebra KX and certain ideals in the so-called letterplace ring KXP has evolved. We continue this research direction, started by La Scala and Levandovskyy, and present novel ideas, supported by the implementation, for effective computations with ideals in the free algebra by utilizing the generalized letterplace correspondance. In particular, we provide a direct algorithm to compute Gröbner bases of non-graded ideals. Surprizingly we realize its behavior as "homogenizing without a homogenization variable". Moreover, we develop new shift-invariant data structures for this family of algorithms and discuss about them. Furthermore we generalize the famous criteria of Gebauer-Möller to the non-commutative setting and show the benefits for the computation by allowing to skip unnecessary critical pairs. The methods are implemented in the computer algebra system Singular. We present a comparison of performance of our implementation with the corresponding implementations in the systems Magma [BCP97] and GAP [GAP13] on the representative set of nontrivial examples.
{"title":"Enhanced computations of gröbner bases in free algebras as a new application of the letterplace paradigm","authors":"V. Levandovskyy, Grischa Studzinski, Benjamin Schnitzler","doi":"10.1145/2465506.2465948","DOIUrl":"https://doi.org/10.1145/2465506.2465948","url":null,"abstract":"Recently, the notion of \"letterplace correspondence\" between ideals in the free associative algebra KX and certain ideals in the so-called letterplace ring KXP has evolved. We continue this research direction, started by La Scala and Levandovskyy, and present novel ideas, supported by the implementation, for effective computations with ideals in the free algebra by utilizing the generalized letterplace correspondance. In particular, we provide a direct algorithm to compute Gröbner bases of non-graded ideals. Surprizingly we realize its behavior as \"homogenizing without a homogenization variable\". Moreover, we develop new shift-invariant data structures for this family of algorithms and discuss about them.\u0000 Furthermore we generalize the famous criteria of Gebauer-Möller to the non-commutative setting and show the benefits for the computation by allowing to skip unnecessary critical pairs. The methods are implemented in the computer algebra system Singular. We present a comparison of performance of our implementation with the corresponding implementations in the systems Magma [BCP97] and GAP [GAP13] on the representative set of nontrivial examples.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115096598","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 propose efficient algorithms to compute the Gröbner basis of an ideal I subset k[x1,...,xn] globally invariant under the action of a commutative matrix group G, in the non-modular case (where char(k) doesn't divide |G|). The idea is to simultaneously diagonalize the matrices in G, and apply a linear change of variables on I corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on I are diagonal. This action induces a grading on the ring R=k[x1,...,xn], compatible with the degree, indexed by a group related to G, that we call G-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into |G| submatrices of roughly the same size. In the same way, we are able to split the canonical basis of R/I (the staircase) if I is a zero-dimensional ideal. Therefore, we derive abelian versions of the classical algorithms F4, F5 or FGLM. Moreover, this new variant of F4/ F5 allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of F4. Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.
{"title":"Gröbner bases of ideals invariant under a commutative group: the non-modular case","authors":"J. Faugère, Jules Svartz","doi":"10.1145/2465506.2465944","DOIUrl":"https://doi.org/10.1145/2465506.2465944","url":null,"abstract":"We propose efficient algorithms to compute the Gröbner basis of an ideal <i>I</i> subset <i>k</i>[<i>x</i><sub>1</sub>,...,<i>x<sub>n</sub></i>] globally invariant under the action of a commutative matrix group <i>G</i>, in the non-modular case (where <i>char</i>(<i>k</i>) doesn't divide |<i>G</i>|). The idea is to simultaneously diagonalize the matrices in <i>G</i>, and apply a linear change of variables on <i>I</i> corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on <i>I</i> are diagonal. This action induces a grading on the ring <i>R=k</i>[<i>x</i><sub>1</sub>,...,<i>x<sub>n</sub></i>], compatible with the degree, indexed by a group related to <i>G</i>, that we call <i>G</i>-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into |<i>G</i>| submatrices of roughly the same size. In the same way, we are able to split the canonical basis of <i>R/I</i> (the staircase) if <i>I</i> is a zero-dimensional ideal. Therefore, we derive <i>abelian</i> versions of the classical algorithms <i>F</i><sub>4</sub>, <i>F</i><sub>5</sub> or FGLM. Moreover, this new variant of <i>F</i><sub>4</sub>/ <i>F</i><sub>5</sub> allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of <i>F</i><sub>4</sub>. Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"236 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125757306","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 give a detailed description of the interface between the MATHEMAGIX language and C++. In particular, we describe the mechanism which allows us to import a C++ template library (which only permits static instantiation) as a fully generic MATHEMAGIX template library.
{"title":"Interfacing mathemagix with C++","authors":"J. Hoeven, Grégoire Lecerf","doi":"10.1145/2465506.2465511","DOIUrl":"https://doi.org/10.1145/2465506.2465511","url":null,"abstract":"In this paper, we give a detailed description of the interface between the MATHEMAGIX language and C++. In particular, we describe the mechanism which allows us to import a C++ template library (which only permits static instantiation) as a fully generic MATHEMAGIX template library.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"43 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133071221","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 proposes an O(n4) quantum Fourier transform (QFT) algorithm over symmetric group Sn, the fastest QFT algorithm of its kind. We propose a fast Fourier transform algorithm over symmetric group Sn, which consists of O(n3) multiplications of unitary matrices, and then transform it into a quantum circuit form. The QFT algorithm can be applied to constructing the standard algorithm of the hidden subgroup problem.
{"title":"Quantum fourier transform over symmetric groups","authors":"Y. Kawano, Hiroshi Sekigawa","doi":"10.1145/2465506.2465940","DOIUrl":"https://doi.org/10.1145/2465506.2465940","url":null,"abstract":"This paper proposes an <i>O</i>(<i>n</i><sup>4</sup>) quantum Fourier transform (QFT) algorithm over symmetric group <i>S<sub>n</sub></i>, the fastest QFT algorithm of its kind. We propose a fast Fourier transform algorithm over symmetric group <i>S<sub>n</sub></i>, which consists of <i>O</i>(<i>n</i><sup>3</sup>) multiplications of unitary matrices, and then transform it into a quantum circuit form. The QFT algorithm can be applied to constructing the standard algorithm of the hidden subgroup problem.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124144026","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 calculus of moving surfaces (CMS) is an analytic framework that extends the tensor calculus to deforming manifolds. We have applied the CMS to a number of boundary variation problems using a Term Rewrite System (TRS). The TRS is used to convert the initial CMS expression into a form that can be evaluated. The CMS produces expressions that are true for all coordinate spaces. This makes it very powerful but applications remain limited by a rapid growth in the size of expressions. We have extended results on existing problems to orders that had been previously intractable. In this paper, we describe our TRS and our method for evaluating CMS expressions on a specific coordinate system. Our work has already provided new insight into problems of current interest to researchers in the CMS.
{"title":"A term rewriting system for the calculus of moving surfaces","authors":"Mark Boady, P. Grinfeld, Jeremy R. Johnson","doi":"10.1145/2465506.2466576","DOIUrl":"https://doi.org/10.1145/2465506.2466576","url":null,"abstract":"The calculus of moving surfaces (CMS) is an analytic framework that extends the tensor calculus to deforming manifolds. We have applied the CMS to a number of boundary variation problems using a Term Rewrite System (TRS). The TRS is used to convert the initial CMS expression into a form that can be evaluated. The CMS produces expressions that are true for all coordinate spaces. This makes it very powerful but applications remain limited by a rapid growth in the size of expressions. We have extended results on existing problems to orders that had been previously intractable. In this paper, we describe our TRS and our method for evaluating CMS expressions on a specific coordinate system. Our work has already provided new insight into problems of current interest to researchers in the CMS.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123413214","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 introduce the RB algorithm for Gröbner basis computation, a simpler yet equivalent algorithm to F5GEN. RB contains the original unmodified F5 algorithm as a special case, so it is possible to study and understand F5 by considering the simpler RB. We present simple yet complete proofs of this fact and of F5's termination and correctness. RB is parametrized by a rewrite order and it contains many published algorithms as special cases, including SB. We prove that SB is the best possible instantiation of RB in the following sense. Let X be any instantiation of RB (such as F5). Then the S-pairs reduced by SB are always a subset of the S-pairs reduced by X and the basis computed by SB is always a subset of the basis computed by X.
{"title":"Signature rewriting in gröbner basis computation","authors":"C. Eder, B. Roune","doi":"10.1145/2465506.2465522","DOIUrl":"https://doi.org/10.1145/2465506.2465522","url":null,"abstract":"We introduce the RB algorithm for Gröbner basis computation, a simpler yet equivalent algorithm to F5GEN. RB contains the original unmodified F5 algorithm as a special case, so it is possible to study and understand F5 by considering the simpler RB. We present simple yet complete proofs of this fact and of F5's termination and correctness. RB is parametrized by a rewrite order and it contains many published algorithms as special cases, including SB.\u0000 We prove that SB is the best possible instantiation of RB in the following sense. Let X be any instantiation of RB (such as F5). Then the S-pairs reduced by SB are always a subset of the S-pairs reduced by X and the basis computed by SB is always a subset of the basis computed by X.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126150931","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 extend complex homotopy methods to finding witness points on the irreducible components of real varieties. In particular we construct such witness points as the isolated real solutions of a constrained optimization problem. First a random hyperplane characterized by its random normal vector is chosen. Witness points are computed by a polyhedral homotopy method. Some of them are at the intersection of this hyperplane with the components. Other witness points are the local critical points of the distance from the plane to components. A method is also given for constructing regular witness points on components, when the critical points are singular. The method is applicable to systems satisfying certain regularity conditions. Illustrative examples are given. We show that the method can be used in the consistent initialization phase of a popular method due to Pryce and Pantelides for preprocessing differential algebraic equations for numerical solution.
{"title":"Finding points on real solution components and applications to differential polynomial systems","authors":"Wenyuan Wu, G. Reid","doi":"10.1145/2465506.2465954","DOIUrl":"https://doi.org/10.1145/2465506.2465954","url":null,"abstract":"In this paper we extend complex homotopy methods to finding witness points on the irreducible components of real varieties. In particular we construct such witness points as the isolated real solutions of a constrained optimization problem.\u0000 First a random hyperplane characterized by its random normal vector is chosen. Witness points are computed by a polyhedral homotopy method. Some of them are at the intersection of this hyperplane with the components. Other witness points are the local critical points of the distance from the plane to components. A method is also given for constructing regular witness points on components, when the critical points are singular.\u0000 The method is applicable to systems satisfying certain regularity conditions. Illustrative examples are given. We show that the method can be used in the consistent initialization phase of a popular method due to Pryce and Pantelides for preprocessing differential algebraic equations for numerical solution.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127090810","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 present a new heuristic algorithm for computing the determinant of a nonsingular n x n integer matrix. Extensive empirical results from a highly optimized implementation show the running time grows approximately as n3 log n, even for input matrices with a highly nontrivial Smith invariant structure. We extend the algorithm to compute the Hermite form of the input matrix. Both the determinant and Hermite form algorithm certify correctness of the computed results.
{"title":"Computing the invariant structure of integer matrices: fast algorithms into practice","authors":"Colton Pauderis, A. Storjohann","doi":"10.1145/2465506.2465955","DOIUrl":"https://doi.org/10.1145/2465506.2465955","url":null,"abstract":"We present a new heuristic algorithm for computing the determinant of a nonsingular n x n integer matrix. Extensive empirical results from a highly optimized implementation show the running time grows approximately as n3 log n, even for input matrices with a highly nontrivial Smith invariant structure. We extend the algorithm to compute the Hermite form of the input matrix. Both the determinant and Hermite form algorithm certify correctness of the computed results.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129790652","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 L be a second order linear homogeneous differential equation with rational function coefficients. The goal in this paper is to solve L in terms of hypergeometric function 2F1(a,b;c|f) where f is a rational function of degree 3.
{"title":"Second order differential equations with hypergeometric solutions of degree three","authors":"V. Kunwar, M. V. Hoeij","doi":"10.1145/2465506.2465953","DOIUrl":"https://doi.org/10.1145/2465506.2465953","url":null,"abstract":"Let L be a second order linear homogeneous differential equation with rational function coefficients. The goal in this paper is to solve L in terms of hypergeometric function 2F1(a,b;c|f) where f is a rational function of degree 3.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123538832","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 assume that a real square-free polynomial A has a degree d, a maximum coefficient bitsize τ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the Double Exponential Sieve algorithm (also called the Bisection of the Exponents), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of t=2-L. The algorithm has Boolean complexity ÕB(d2 τ + d L ). Our algorithms support the same complexity bound for the refinement of r roots, for any r ≤ d.
我们假设一个实数无平方多项式a的度数为d，最大系数位大小为τ，实数根位于隔离区间内，附近没有非实数根(我们量化了这个假设)。然后，我们将双指数筛算法(也称为指数的二分法)、二分法和牛顿迭代结合起来，将该包含区间的宽度减小t=2-L。该算法具有布尔复杂度ÕB(d2 τ + d L)。对于任何r≤d，我们的算法对r根的细化支持相同的复杂度界。
{"title":"On the boolean complexity of real root refinement","authors":"V. Pan, Elias P. Tsigaridas","doi":"10.1145/2465506.2465938","DOIUrl":"https://doi.org/10.1145/2465506.2465938","url":null,"abstract":"We assume that a real square-free polynomial <i>A</i> has a degree <i>d</i>, a maximum coefficient bitsize τ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the <i>Double Exponential Sieve</i> algorithm (also called the <i>Bisection of the Exponents</i>), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of <i>t</i>=2<sup>-L</sup>. The algorithm has Boolean complexity Õ<sub>B</sub>(d<sup>2</sup> τ + d L ). Our algorithms support the same complexity bound for the refinement of <i>r</i> roots, for any <i>r ≤ d</i>.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125467084","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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