In fond memory of Marko Petkovšek (1955-2023), a great summer and enumerator.
怀念马尔科Petkovšek(1955-2023)，一个伟大的夏季和枚举者。
{"title":"Counting Clean Words According to the Number of Their Clean Neighbors","authors":"S. B. Ekhad, D. Zeilberger","doi":"10.1145/3610377.3610379","DOIUrl":"https://doi.org/10.1145/3610377.3610379","url":null,"abstract":"In fond memory of Marko Petkovšek (1955-2023), a great summer and enumerator.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"57 1","pages":"5 - 9"},"PeriodicalIF":0.1,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42327359","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 notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and sufficient condition) for the infinite-dimensionality of its space VL of solutions belonging to R is the presence of a lacunary sequence in VL.
{"title":"Linear Difference Operators with Sequence Coefficients Having Infinite-Dimentional Solution Spaces","authors":"S. Abramov, G. Pogudin","doi":"10.1145/3610377.3610378","DOIUrl":"https://doi.org/10.1145/3610377.3610378","url":null,"abstract":"The notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and sufficient condition) for the infinite-dimensionality of its space VL of solutions belonging to R is the presence of a lacunary sequence in VL.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"57 1","pages":"1 - 4"},"PeriodicalIF":0.1,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44068516","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}
During the last months of his life, while still undergoing chemotherapy after a recent severe oncological operation, Marko remained very much engaged in his work. Actually, not only did he collaborate with one of his young students on an article which was later accepted for publication. He co-wrote as well another one on the subject of the representation of prime numbers by quadratic forms with a now deceased surveyor adept at computer experiments. Marko both came up with the formulation and proof of the corresponding hypotheses.
{"title":"Marko Petkovšek 1955--2023","authors":"Eva U. Petkovšek","doi":"10.1145/3610377.3610380","DOIUrl":"https://doi.org/10.1145/3610377.3610380","url":null,"abstract":"During the last months of his life, while still undergoing chemotherapy after a recent severe oncological operation, Marko remained very much engaged in his work. Actually, not only did he collaborate with one of his young students on an article which was later accepted for publication. He co-wrote as well another one on the subject of the representation of prime numbers by quadratic forms with a now deceased surveyor adept at computer experiments. Marko both came up with the formulation and proof of the corresponding hypotheses.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"57 1","pages":"10 - 18"},"PeriodicalIF":0.1,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48330624","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 dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.
{"title":"Algorithms for Quadratic Forms over Global Function Fields","authors":"Mawunyo Kofi Darkey-Mensah","doi":"10.1145/3609983.3609985","DOIUrl":"https://doi.org/10.1145/3609983.3609985","url":null,"abstract":"This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"150 - 150"},"PeriodicalIF":0.1,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42574074","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 give a computer-free proof that J2 is isomorphic to the progenitor 2*32 : (21+4 : A5) factored by two relations, one of length 3 and and one of length 6, in the symmetric generators.
{"title":"Symmetric Generation of J2 on 32 Letters","authors":"Connie Corona, Zahid Hasan, Bronson Lim","doi":"10.1145/3609983.3609984","DOIUrl":"https://doi.org/10.1145/3609983.3609984","url":null,"abstract":"We give a computer-free proof that J2 is isomorphic to the progenitor 2*32 : (21+4 : A5) factored by two relations, one of length 3 and and one of length 6, in the symmetric generators.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"133 - 149"},"PeriodicalIF":0.1,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41759116","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 current implementation of Thiele rational interpolation in Maple (the Thieleinterpolation routine) breaks down when the points are not well-ordered. In this article, it is shown how this breakdown can be avoided by ordering the interpolation points in an adaptive way.
{"title":"Adaptive Thiele interpolation","authors":"O. S. Celis","doi":"10.1145/3594252.3594254","DOIUrl":"https://doi.org/10.1145/3594252.3594254","url":null,"abstract":"The current implementation of Thiele rational interpolation in Maple (the Thieleinterpolation routine) breaks down when the points are not well-ordered. In this article, it is shown how this breakdown can be avoided by ordering the interpolation points in an adaptive way.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"125 - 132"},"PeriodicalIF":0.1,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43969590","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 article surveys a very new method of enhancing Buchberger's Gröbner basis algorithm by the PRSs (polynomial remainder sequences) and the GCDs of multivariate polynomials. Let F = {F1,...,Fm+1} ⊂ K[x, u] be a given system, where (x) = (x1,...,xm) and (u) = (u1,...,un). Currently, we treat only such Fs that are "healthy" (see the text). Let [EQUATION], where [EQUATION], be the reduced Gröbner basis of ideal (F) w.r.t. the lexicographic order, to be abbreviated to GB(F). Let [EQUATION], be such that [EQUATION] is a small multiple of G1, and the leading monomial of [EQUATION], is a multiple (hopefully small) of the leading monomial of Gi. Our method computes [EQUATION] first, then computes [EQUATION]. Finally, we will apply Buchberger's method to system [EQUATION]. Four new theorems are given. The first and second ones are to compute the lowest-order element of the ideal generated by relatively prime G, H ∈ K[x, u], without computing any Spolynomial. The third theorem says that if F is healthy then [EQUATION]. We compute resultants in K[u], of F through different routes. Then, by Theorem 3, the resultants will be different multiples of G1. Hence, the GCD of them will be a small multiple of G1. In the elimination of x through different routes, we obtain sets of similar remainders such that the elements of each set have the same leading variable and nearly the same degrees. We call the leading coefficients of mutually similar remainders an "LCsystem". We eliminate the leading variables of suitably chosen LCsystems. The fourth theorem constructs a polynomial [EQUATION], such that the leading coefficient of [EQUATION] is the GCD of resultants of elements of an LCsystem chosen.
{"title":"A Bridge between Euclid and Buchberger: (An Attempt to Enhance Gröbner Basis Algorithm by PRSs and GCDs)","authors":"Tateaki Sasaki","doi":"10.1145/3594252.3594253","DOIUrl":"https://doi.org/10.1145/3594252.3594253","url":null,"abstract":"This article surveys a very new method of enhancing Buchberger's Gröbner basis algorithm by the PRSs (polynomial remainder sequences) and the GCDs of multivariate polynomials. Let F = {F1,...,Fm+1} ⊂ K[x, u] be a given system, where (x) = (x1,...,xm) and (u) = (u1,...,un). Currently, we treat only such Fs that are \"healthy\" (see the text). Let [EQUATION], where [EQUATION], be the reduced Gröbner basis of ideal (F) w.r.t. the lexicographic order, to be abbreviated to GB(F). Let [EQUATION], be such that [EQUATION] is a small multiple of G1, and the leading monomial of [EQUATION], is a multiple (hopefully small) of the leading monomial of Gi. Our method computes [EQUATION] first, then computes [EQUATION]. Finally, we will apply Buchberger's method to system [EQUATION]. Four new theorems are given. The first and second ones are to compute the lowest-order element of the ideal generated by relatively prime G, H ∈ K[x, u], without computing any Spolynomial. The third theorem says that if F is healthy then [EQUATION]. We compute resultants in K[u], of F through different routes. Then, by Theorem 3, the resultants will be different multiples of G1. Hence, the GCD of them will be a small multiple of G1. In the elimination of x through different routes, we obtain sets of similar remainders such that the elements of each set have the same leading variable and nearly the same degrees. We call the leading coefficients of mutually similar remainders an \"LCsystem\". We eliminate the leading variables of suitably chosen LCsystems. The fourth theorem constructs a polynomial [EQUATION], such that the leading coefficient of [EQUATION] is the GCD of resultants of elements of an LCsystem chosen.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"97 - 124"},"PeriodicalIF":0.1,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45433243","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 report on work-in-progress to create an SMT-solver inside Maple for non-linear real arithmetic (NRA). We give background information on the algorithm being implemented: cylindrical algebraic coverings as a theory solver in the lazy SMT paradigm. We then present some new work on the identification of minimal conflicting cores from the coverings.
{"title":"An SMT solver for non-linear real arithmetic inside maple","authors":"AmirHosein Sadeghimanesh, M. England","doi":"10.1145/3572867.3572880","DOIUrl":"https://doi.org/10.1145/3572867.3572880","url":null,"abstract":"We report on work-in-progress to create an SMT-solver inside Maple for non-linear real arithmetic (NRA). We give background information on the algorithm being implemented: cylindrical algebraic coverings as a theory solver in the lazy SMT paradigm. We then present some new work on the identification of minimal conflicting cores from the coverings.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"76 - 79"},"PeriodicalIF":0.1,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46776264","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}
Pari/GP was designed for algebraic number theory. As such it does not have some standard features that other general purpose computer algebra systems have such as Gröbner bases and multivariate polynomial factorization. In this letter I say a few words about the history of Pari/GP, the impact Pari/GP has had on number theory, and some of the unique software design features of Pari/GP. I've also included a short bio from Henri, Bill and Karim.
{"title":"Jenks prize announcement","authors":"H. Cohen, Bill Allombert, K. Belabas","doi":"10.1145/3572867.3572884","DOIUrl":"https://doi.org/10.1145/3572867.3572884","url":null,"abstract":"Pari/GP was designed for algebraic number theory. As such it does not have some standard features that other general purpose computer algebra systems have such as Gröbner bases and multivariate polynomial factorization. In this letter I say a few words about the history of Pari/GP, the impact Pari/GP has had on number theory, and some of the unique software design features of Pari/GP. I've also included a short bio from Henri, Bill and Karim.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"92 - 94"},"PeriodicalIF":0.1,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42922689","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}
Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[x1,..., xn] over K and a finite sequence of polynomials f1,...,fc in R with c ≤ n. Let V ⊂ Kn be the algebraic set defined by the simultaneous vanishing of the fi's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set Vc which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f1,...,fc.
{"title":"Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system","authors":"C. Eder, Pierre Lairez, Rafael Mohr, M. S. E. Din","doi":"10.1145/3572867.3572872","DOIUrl":"https://doi.org/10.1145/3572867.3572872","url":null,"abstract":"Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[x1,..., xn] over K and a finite sequence of polynomials f1,...,fc in R with c ≤ n. Let V ⊂ Kn be the algebraic set defined by the simultaneous vanishing of the fi's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set Vc which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f1,...,fc.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"56 1","pages":"41 - 45"},"PeriodicalIF":0.1,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43770067","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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