We aim to factor a sparse polynomial a ∈ Z[x1, ···,xn] represented by a black box. The authors have previously developed efficient sparse Hensel lifting algorithms for the monic and square-free case that outperforms the algorithm by Kaltofen and Trager in 1990. We complete this black box factorization problem for the non-monic case with a new algorithm that computes the factors of a using many non-monic bivariate Hensel lifts. Our algorithm handles all cases of input a ∈ Z[x1, ···,xn] including the non-square-free and the non-primitive cases. We have implemented the algorithm in Maple with all major subroutines coded in C for efficiency.
{"title":"Factoring non-monic polynomials represented by black boxes","authors":"Tian Chen, M. Monagan","doi":"10.1145/3572867.3572881","DOIUrl":"https://doi.org/10.1145/3572867.3572881","url":null,"abstract":"We aim to factor a sparse polynomial a ∈ Z[x1, ···,xn] represented by a black box. The authors have previously developed efficient sparse Hensel lifting algorithms for the monic and square-free case that outperforms the algorithm by Kaltofen and Trager in 1990. We complete this black box factorization problem for the non-monic case with a new algorithm that computes the factors of a using many non-monic bivariate Hensel lifts. Our algorithm handles all cases of input a ∈ Z[x1, ···,xn] including the non-square-free and the non-primitive cases. We have implemented the algorithm in Maple with all major subroutines coded in C for efficiency.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"80 - 83"},"PeriodicalIF":0.1,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46100619","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 an identity on multinomial coefficients, as well as the proof for it.
我们介绍了多项式系数的一个恒等式，以及它的证明。
{"title":"An identity on multinomial coefficients","authors":"Jiayue Qi","doi":"10.1145/3572867.3572878","DOIUrl":"https://doi.org/10.1145/3572867.3572878","url":null,"abstract":"We introduce an identity on multinomial coefficients, as well as the proof for it.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"68 - 71"},"PeriodicalIF":0.1,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47689819","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 an algorithm for factoring linear differential operators with coefficients in a finite separable extension of Fp(x). Our methods rely on specific tools arising in positive characteristic: p-curvature, structure of simple central algebras and p-Riccati equations.
{"title":"Factoring differential operators over algebraic curves in positive characteristic","authors":"Raphaël Pagès","doi":"10.1145/3572867.3572876","DOIUrl":"https://doi.org/10.1145/3572867.3572876","url":null,"abstract":"We present an algorithm for factoring linear differential operators with coefficients in a finite separable extension of Fp(x). Our methods rely on specific tools arising in positive characteristic: p-curvature, structure of simple central algebras and p-Riccati equations.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"60 - 63"},"PeriodicalIF":0.1,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44952706","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 Mathematica tools for proving or disproving whether a set of objects constitutes a vector space. When necessary axioms are upheld, the relationships between the variables are presented. When the axioms fail, intuitive counterexamples are shown. A number of different kinds of vectors are demonstrated, with varying types of vector addition and scalar multiplication as well. All of the calculations are performed in an automated fashion.
{"title":"Automated vector space proofs using mathematica","authors":"Aaron E. Naiman","doi":"10.1145/3572865.3572866","DOIUrl":"https://doi.org/10.1145/3572865.3572866","url":null,"abstract":"We present Mathematica tools for proving or disproving whether a set of objects constitutes a vector space. When necessary axioms are upheld, the relationships between the variables are presented. When the axioms fail, intuitive counterexamples are shown. A number of different kinds of vectors are demonstrated, with varying types of vector addition and scalar multiplication as well. All of the calculations are performed in an automated fashion.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"1 - 13"},"PeriodicalIF":0.1,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41786055","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}
Shahriar Iravanian, Carl Martensen, Alessandro Cheli, Shashi Gowda, Anand Jain, Yingbo Ma, Chris Rackauckas
The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.
{"title":"Symbolic-numeric integration of univariate expressions based on sparse regression","authors":"Shahriar Iravanian, Carl Martensen, Alessandro Cheli, Shashi Gowda, Anand Jain, Yingbo Ma, Chris Rackauckas","doi":"10.1145/3572867.3572882","DOIUrl":"https://doi.org/10.1145/3572867.3572882","url":null,"abstract":"The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"84 - 87"},"PeriodicalIF":0.1,"publicationDate":"2022-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44193914","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 RISC1 curriculum is open to Ph.D. students with a background in either mathematics or computer science and a strong interest (and preferably prior knowledge) in the respective other area.
RISC1课程向具有数学或计算机科学背景、对各自其他领域有浓厚兴趣（最好是先验知识）的博士生开放。
{"title":"RISC Ph.D. studies program","authors":"Ralf Hemmecke","doi":"10.1145/3551872.3551874","DOIUrl":"https://doi.org/10.1145/3551872.3551874","url":null,"abstract":"The RISC1 curriculum is open to Ph.D. students with a background in either mathematics or computer science and a strong interest (and preferably prior knowledge) in the respective other area.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"135 - 135"},"PeriodicalIF":0.1,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43098962","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}
Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA.
{"title":"Abstracts of recent doctoral dissertations in computer algebra","authors":"Cca editors","doi":"10.1145/3511528.3511540","DOIUrl":"https://doi.org/10.1145/3511528.3511540","url":null,"abstract":"Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"117 - 124"},"PeriodicalIF":0.1,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41969286","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 devoted to present a new method for computing the approximate Greatest Common Divisor (GCD) of several polynomials (not pairwise) from the generalized Hankel matrix. Our approach based on the calculation of cofactors is tested for several sets of polynomials.
{"title":"Approximate greatest common divisor of several polynomials from Hankel matrices","authors":"S. Belhaj, Abdulrahman Alsulami","doi":"10.1145/3511528.3511532","DOIUrl":"https://doi.org/10.1145/3511528.3511532","url":null,"abstract":"This paper is devoted to present a new method for computing the approximate Greatest Common Divisor (GCD) of several polynomials (not pairwise) from the generalized Hankel matrix. Our approach based on the calculation of cofactors is tested for several sets of polynomials.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"77 - 81"},"PeriodicalIF":0.1,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48417071","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 a better algorithm for approximate greatest common divisor (approximate GCD) of univariate polynomials in terms of robustness and distance, based on the NewtonSLRA algorithm that is a solver for the structured low rank approximation (SLRA) problem. Our algorithm mainly enlarges the tangent space in the NewtonSLRA algorithm and adapts it to a certain weighted Frobenius norm. Moreover, we propose some improvement in computing time.
{"title":"Approximate GCD by relaxed NewtonSLRA algorithm","authors":"Kosaku Nagasaka","doi":"10.1145/3511528.3511536","DOIUrl":"https://doi.org/10.1145/3511528.3511536","url":null,"abstract":"We propose a better algorithm for approximate greatest common divisor (approximate GCD) of univariate polynomials in terms of robustness and distance, based on the NewtonSLRA algorithm that is a solver for the structured low rank approximation (SLRA) problem. Our algorithm mainly enlarges the tangent space in the NewtonSLRA algorithm and adapts it to a certain weighted Frobenius norm. Moreover, we propose some improvement in computing time.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"97 - 101"},"PeriodicalIF":0.1,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46680319","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}
Daniel Miguel, Andrea Guidolin, A. Romero, J. Rubio
In this work we present an ongoing project on the development and study of new spectral systems which combine filtrations associated to Serre and Eilenberg-Moore spectral sequences of different fibrations. Our new spectral systems are part of a new module for the Kenzo system and can be useful to deduce new relations on the initial spectral sequences and to obtain information about different filtrations of the homology groups of the fiber and the base space of the fibrations.
{"title":"Constructing new spectral systems from simplicial fibrations","authors":"Daniel Miguel, Andrea Guidolin, A. Romero, J. Rubio","doi":"10.1145/3511528.3511534","DOIUrl":"https://doi.org/10.1145/3511528.3511534","url":null,"abstract":"In this work we present an ongoing project on the development and study of new spectral systems which combine filtrations associated to Serre and Eilenberg-Moore spectral sequences of different fibrations. Our new spectral systems are part of a new module for the Kenzo system and can be useful to deduce new relations on the initial spectral sequences and to obtain information about different filtrations of the homology groups of the fiber and the base space of the fibrations.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"87 - 91"},"PeriodicalIF":0.1,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43837553","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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