Let A, B ∈ K[X,Y] be two bivariate polynomials over an effective field K, and let G be the reduced Gröbner basis of the ideal I := ⟨A, B⟩ generated by A and B with respect to the usual degree lexicographic order. Assuming A and B sufficiently generic, G admits a so-called concise representation that helps computing normal forms more efficiently [7]. Actually, given this concise representation, a polynomial P ∈ K[X, Y] can be reduced modulo G with quasi-optimal complexity (in terms of the size of the input A, B, P). Moreover, the concise representation can be computed from the input A, B with quasi-optimal complexity as well. The present paper reports on an efficient implementation for these two tasks in the free software Mathemagix [10]. This implementation is included in Mathemagix as a library called Larrix.
{"title":"Computing generic bivariate Gröbner bases with Mathemagix","authors":"Robin Larrieu","doi":"10.1145/3371991.3371994","DOIUrl":"https://doi.org/10.1145/3371991.3371994","url":null,"abstract":"Let <i>A, B</i> ∈ K[<i>X,Y</i>] be two bivariate polynomials over an effective field K, and let <i>G</i> be the reduced Gröbner basis of the ideal <i>I</i> := ⟨<i>A, B</i>⟩ generated by <i>A</i> and <i>B</i> with respect to the usual degree lexicographic order. Assuming <i>A</i> and <i>B</i> sufficiently generic, <i>G</i> admits a so-called <i>concise representation</i> that helps computing normal forms more efficiently [7]. Actually, given this concise representation, a polynomial <i>P</i> ∈ K[<i>X, Y</i>] can be reduced modulo <i>G</i> with quasi-optimal complexity (in terms of the size of the input <i>A, B, P</i>). Moreover, the concise representation can be computed from the input <i>A, B</i> with quasi-optimal complexity as well. The present paper reports on an efficient implementation for these two tasks in the free software Mathemagix [10]. This implementation is included in Mathemagix as a library called Larrix.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"530 1","pages":"41-44"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78873559","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}
Implicit is a package for implicitizing rational planar curves and rational tensor product surfaces, developed on Maplesoft based on the state-of-the-art implicitization techniques. The main functions ImpCurve and ImpSurface return the implicit equation of a rational planar curve or a rational tensor product surface. Other popularly used functions, such as ImpDegree, ImpMatrix and ImpRuled that are used for deciding the implicit degree and the implicit matrix of a general rational surface, and computing the implicit equation of a rational ruled surface in a more efficient way, are also proposed.
{"title":"ISSAC 2019 software presentations communicated by Yue Ren: a package to compute implicit equations for rational curves and surfaces","authors":"Shanshan Yao, Yifei Feng, Xiaohong Jia, L. Shen","doi":"10.1145/3371991.3371992","DOIUrl":"https://doi.org/10.1145/3371991.3371992","url":null,"abstract":"Implicit is a package for implicitizing rational planar curves and rational tensor product surfaces, developed on Maplesoft based on the state-of-the-art implicitization techniques. The main functions ImpCurve and ImpSurface return the implicit equation of a rational planar curve or a rational tensor product surface. Other popularly used functions, such as ImpDegree, ImpMatrix and ImpRuled that are used for deciding the implicit degree and the implicit matrix of a general rational surface, and computing the implicit equation of a rational ruled surface in a more efficient way, are also proposed.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"73 1","pages":"33-36"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87995423","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 the latest update of the ore_algebra package for SageMath. The main new feature in this release is the support of operators in several variables.
{"title":"Multivariate ore polynomials in SageMath","authors":"Manuel Kauers, M. Mezzarobba","doi":"10.1145/3371991.3371998","DOIUrl":"https://doi.org/10.1145/3371991.3371998","url":null,"abstract":"We present the latest update of the ore_algebra package for SageMath. The main new feature in this release is the support of operators in several variables.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"52 1","pages":"57-60"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74254822","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}
Matrices or linear operators and their identities can be modelled algebraically by noncommutative polynomials in the free algebra. For proving new identities of matrices or operators from given ones, computations are done formally with noncommutative polynomials. Computations in the free algebra, however, are not necessarily compatible with formats of matrices resp. with domains and codomains of operators. For ensuring validity of such computations in terms of operators, in principle, one would have to inspect every step of the computation. In [9], an algebraic framework is developed that allows to rigorously justify such computations without restricting the computation to compatible expressions. The main result of that paper reduces the proof of an operator identity to verifying membership of the corresponding polynomial in the ideal generated by the polynomials corresponding to the assumptions and verifying compatibility of this polynomial and of the generators of the ideal.
{"title":"Certifying operator identities via noncommutative Gröbner bases","authors":"Clemens Hofstadler, C. Raab, G. Regensburger","doi":"10.1145/3371991.3371996","DOIUrl":"https://doi.org/10.1145/3371991.3371996","url":null,"abstract":"Matrices or linear operators and their identities can be modelled algebraically by noncommutative polynomials in the free algebra. For proving new identities of matrices or operators from given ones, computations are done formally with noncommutative polynomials. Computations in the free algebra, however, are not necessarily compatible with formats of matrices resp. with domains and codomains of operators. For ensuring validity of such computations in terms of operators, in principle, one would have to inspect every step of the computation. In [9], an algebraic framework is developed that allows to rigorously justify such computations without restricting the computation to compatible expressions. The main result of that paper reduces the proof of an operator identity to verifying membership of the corresponding polynomial in the ideal generated by the polynomials corresponding to the assumptions and verifying compatibility of this polynomial and of the generators of the ideal.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"12 1","pages":"49-52"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75107953","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 here the Sage package dd_functions which provides many features to compute with DD-finite functions, a natural extension of the class of holonomic or D-finite functions. The package, focused on a functional approach, allows the user to compute closure properties, extract coefficient sequences and compute the composition (as formal power series) of functions treated in this package. All these operations reduce the problem to linear algebra computations where classical division-free algorithms are used.
{"title":"A sage implementation for DD-finite functions","authors":"Antonio Jiménez-Pastor","doi":"10.1145/3371991.3371997","DOIUrl":"https://doi.org/10.1145/3371991.3371997","url":null,"abstract":"We present here the Sage package dd_functions which provides many features to compute with DD-finite functions, a natural extension of the class of holonomic or D-finite functions. The package, focused on a functional approach, allows the user to compute closure properties, extract coefficient sequences and compute the composition (as formal power series) of functions treated in this package. All these operations reduce the problem to linear algebra computations where classical division-free algorithms are used.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"45 1","pages":"53-56"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88078783","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}
S. Abramov, A. Ryabenko, L. Sevastianov, Min Wu, Yvette Zonn
The International Conference "Computer Algebra"was held in Moscow, Russia from June 17-21, 2019. The conference web-site is http://www.ccas.ru/ca/conference. Co-organized by the Dorodnicyn Computing Centre (Federal Research Center "Computer Science and Control") of Russian Academy of Sciences and the Peoples' Friendship University of Russia, it was devoted to computer algebra and related topics. This is the third edition of the conference, and the previous two were in 2016 and 2017, respectively.
{"title":"The conference \"computer algebra\" in Moscow","authors":"S. Abramov, A. Ryabenko, L. Sevastianov, Min Wu, Yvette Zonn","doi":"10.1145/3371991.3372000","DOIUrl":"https://doi.org/10.1145/3371991.3372000","url":null,"abstract":"The International Conference \"Computer Algebra\"was held in Moscow, Russia from June 17-21, 2019. The conference web-site is http://www.ccas.ru/ca/conference. Co-organized by the Dorodnicyn Computing Centre (Federal Research Center \"Computer Science and Control\") of Russian Academy of Sciences and the Peoples' Friendship University of Russia, it was devoted to computer algebra and related topics.\u0000 This is the third edition of the conference, and the previous two were in 2016 and 2017, respectively.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"53 1","pages":"65-77"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78385452","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}
Julián Cuevas-Rozo, Jose Divasón, Miguel Ángel Marco Buzunáriz, A. Romero
In this work we present an interface between SageMath and Kenzo, together with an optional Kenzo package. Our work makes it possible to communicate both computer algebra programs and enhances the SageMath system with new capabilities in algebraic topology, dealing in particular with simplicial objects of infinite nature.
{"title":"A Kenzo interface for algebraic topology computations in SageMath","authors":"Julián Cuevas-Rozo, Jose Divasón, Miguel Ángel Marco Buzunáriz, A. Romero","doi":"10.1145/3371991.3371999","DOIUrl":"https://doi.org/10.1145/3371991.3371999","url":null,"abstract":"In this work we present an interface between SageMath and Kenzo, together with an optional Kenzo package. Our work makes it possible to communicate both computer algebra programs and enhances the SageMath system with new capabilities in algebraic topology, dealing in particular with simplicial objects of infinite nature.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"26 1","pages":"61-64"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77798439","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 the Macaulay2 package NumericalCertification for certifying roots of square polynomial systems. It employs the interval Krawczyk method and α-theory as main methods for certification. The package works with output data computed in Macaulay2 with no need for external software. Also, our implementation supports the Krawczyk method which uses interval arithmetic.
{"title":"Certifying approximate solutions to polynomial systems on Macaulay2","authors":"Kisun Lee","doi":"10.1145/3371991.3371995","DOIUrl":"https://doi.org/10.1145/3371991.3371995","url":null,"abstract":"We present the Macaulay2 package NumericalCertification for certifying roots of square polynomial systems. It employs the interval Krawczyk method and α-theory as main methods for certification. The package works with output data computed in Macaulay2 with no need for external software. Also, our implementation supports the Krawczyk method which uses interval arithmetic.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"11 1","pages":"45-48"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86712105","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}
It is well known that if the leading matrix of a linear ordinary differential or difference system is nonsingular, then the determinant of this matrix contains some useful information on solutions of the system. We investigate a kind of non-arithmetic complexity of known algorithms for transforming a matrix of scalar operators to an equivalent matrix which has non-singular frontal, or, leading matrix. In the algorithms under consideration, the differentiation in the differential case and the shift in the difference case play a significant role. We give some analysis of the complexity measured as the number of differentiations or, resp., shifts in the worst case. We not only offer estimates of the complexity written using the O-notation, but we also show that some estimates are sharp and can not be improved.
{"title":"Row reduction process for matrices of scalar operators: storing the intermediate results of row transformation","authors":"S. Abramov, M. Barkatou","doi":"10.1145/3363520.3363522","DOIUrl":"https://doi.org/10.1145/3363520.3363522","url":null,"abstract":"It is well known that if the leading matrix of a linear ordinary differential or difference system is nonsingular, then the determinant of this matrix contains some useful information on solutions of the system. We investigate a kind of non-arithmetic complexity of known algorithms for transforming a matrix of scalar operators to an equivalent matrix which has non-singular frontal, or, leading matrix. In the algorithms under consideration, the differentiation in the differential case and the shift in the difference case play a significant role. We give some analysis of the complexity measured as the number of differentiations or, resp., shifts in the worst case. We not only offer estimates of the complexity written using the O-notation, but we also show that some estimates are sharp and can not be improved.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"20 1","pages":"23-30"},"PeriodicalIF":0.0,"publicationDate":"2019-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78591360","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 work presented here is motivated by our ongoing project on symbolic bifurcation analysis of multidimensional germs of singularities, say [EQUATION] where z and λ denote the state variables and a distinguished parameter, respectively. A local solution Z(λ) for f(Z(λ), λ) = 0 is called a bifurcation diagram. Any two bifurcation diagrams Z1(λ) and Z2(λ) of f (z, λ) are called qualitatively equivalent when there exists a diffeomorphism (Φ(Z1, λ1), Λ(λ1)) transforming the bifurcation diagram Z1(λ1) into Z2(λ2) = Φ(Z1(Λ-1(lD2)), Λ-1(λ2)). Our proposed bifurcation analysis often corresponds to steady state bifurcations of PDEs, static and dynamical systems and occur in many real life and engineering control problems [5]. This will be integrated into the Singularity library [3, 4] as MultiDimSingularities module. Our library provides systematic tools for symbolic treatment of bifurcation analysis of such systems. The classical approach uses root finding and parametric continuation methods through numerical normal form analysis for differential systems with low codimension (degeneracy). The latter fails on comprehensive analysis for a parametric system and for systems with moderate degeneracies. The goal here is to classify the qualitatively different bifurcation diagrams of an unfolding germ [EQUATION], for f (z, λ), where α = (α1,...,αp) are unfolding parameters and F(z, λ, 0) = f(z, λ). This classification is feasible via the concept of persistent bifurcation diagrams. A diagram is called persistent when it is qualitatively self-equivalent under arbitrarily small perturbations. The idea is to find non-persistent sources of bifurcations. Non-persistent sources are divided into three categories: bifurcation B, hysteresis H and double limit point D; see [6, Page 410]. The set [EQUATION] is called a transition local variety, that is a codimension-1 hyperplane in the parameter space. A local variety refers to a neighborhood subset of zeros of a polynomial system. The complement space of this hyperplane consists of a finite number of connected components, say C1, ..., Cn. For any i and α, β ∈ Ci, the bifurcation diagrams Z(λ, α) and Z(λ, β) are qualitatively equivalent. Hence, a choice of bifurcation diagram from each connected component gives rise to a complete list of persistent bifurcation diagram classifications for F(z, λ, α). The RegularChains library in Maple provides a tool through the command CylindricalAlgebraicDecompose enabling us to generate this list; see [1]. However, there usually exist mu
{"title":"Gröbner bases and multi-dimensional persistent bifurcation diagram classifications","authors":"M. Gazor, A. Hashemi, Mahsa Kazemi","doi":"10.1145/3338637.3338640","DOIUrl":"https://doi.org/10.1145/3338637.3338640","url":null,"abstract":"The work presented here is motivated by our ongoing project on symbolic bifurcation analysis of multidimensional germs of singularities, say\u0000 [EQUATION]\u0000 where <i>z</i> and λ denote the state variables and a distinguished parameter, respectively. A local solution <i>Z</i>(λ) for <i>f</i>(<i>Z</i>(λ), λ) = 0 is called a <i>bifurcation diagram.</i> Any two bifurcation diagrams <i>Z</i><sub>1</sub>(λ) and <i>Z</i><sub>2</sub>(λ) of <i>f</i> (<i>z</i>, λ) are called qualitatively equivalent when there exists a diffeomorphism (Φ(<i>Z</i><sub>1</sub>, λ<sub>1</sub>), Λ(λ<sub>1</sub>)) transforming the bifurcation diagram <i>Z</i><sub>1</sub>(λ<sub>1</sub>) into <i>Z</i><sub>2</sub>(λ<sub>2</sub>) = Φ(<i>Z</i><sub>1</sub>(Λ<sup>-1</sup>(lD<sub>2</sub>)), Λ<sup>-1</sup>(λ<sub>2</sub>)). Our proposed bifurcation analysis often corresponds to steady state bifurcations of PDEs, static and dynamical systems and occur in many real life and engineering control problems [5]. This will be integrated into the Singularity library [3, 4] as MultiDimSingularities module. Our library provides systematic tools for symbolic treatment of bifurcation analysis of such systems. The classical approach uses root finding and parametric continuation methods through numerical normal form analysis for differential systems with low codimension (degeneracy). The latter fails on comprehensive analysis for a parametric system and for systems with moderate degeneracies. The goal here is to classify the qualitatively different bifurcation diagrams of an <i>unfolding</i> germ\u0000 [EQUATION],\u0000 for <i>f</i> (<i>z</i>, λ), where α = (α<sub>1</sub>,...,α<sub><i>p</i></sub>) are unfolding parameters and <i>F</i>(<i>z</i>, λ, 0) = <i>f</i>(<i>z</i>, λ). This classification is feasible via the concept of persistent bifurcation diagrams. A diagram is called <i>persistent</i> when it is qualitatively self-equivalent under arbitrarily small perturbations. The idea is to find <i>non-persistent</i> sources of bifurcations. Non-persistent sources are divided into three categories: <i>bifurcation B, hysteresis H and double limit point D</i>; see [6, Page 410]. The set [EQUATION] is called a <i>transition local variety</i>, that is a codimension-1 hyperplane in the parameter space. A <i>local variety</i> refers to a neighborhood subset of zeros of a polynomial system. The complement space of this hyperplane consists of a finite number of connected components, say <i>C</i><sub>1</sub>, ..., <i>C<sub>n</sub></i>. For any <i>i</i> and α, β ∈ <i>C<sub>i</sub></i>, the bifurcation diagrams <i>Z</i>(λ, α) and <i>Z</i>(λ, β) are qualitatively equivalent. Hence, a choice of bifurcation diagram from each connected component gives rise to a complete list of <i>persistent bifurcation diagram classifications</i> for <i>F</i>(<i>z</i>, λ, α). The RegularChains library in Maple provides a tool through the command CylindricalAlgebraicDecompose enabling us to generate this list; see [1]. However, there usually exist mu","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"85 1","pages":"120-122"},"PeriodicalIF":0.0,"publicationDate":"2019-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74583177","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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