{"title":"Recursive structures in involutive bases theory","authors":"Amir Hashemi , Matthias Orth , Werner M. Seiler","doi":"10.1016/j.jsc.2023.01.003","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.01.003","url":null,"abstract":"<div><p>We study characterisations of involutive bases using a recursion over the variables in the underlying polynomial ring and corresponding completion algorithms. Three key ingredients are (i) an old result by Janet recursively characterising Janet bases for which we provide a new and simpler proof, (ii) the Berkesch–Schreyer variant of Buchberger's algorithm and (iii) a tree representation of sets of terms also known as Janet trees. We start by extending Janet's result to a recursive criterion for minimal Janet bases leading to an algorithm to minimise any given Janet basis. We then extend Janet's result also to Janet-like bases as introduced by Gerdt and Blinkov. Next, we design a novel recursive completion algorithm for Janet bases. We study then the extension of these results to Pommaret bases. It yields a novel recursive characterisation of quasi-stability which we use for deterministically constructing “good” coordinates more efficiently than in previous works. A small modification leads to a novel deterministic algorithm for putting an ideal into Nœther position. Finally, we provide a general theory of involutive-like bases with special emphasis on Pommaret-like bases and study the syzygy theory of Janet-like and Pommaret-like bases.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"MacWilliams' Extension Theorem for rank-metric codes","authors":"Elisa Gorla, Flavio Salizzoni","doi":"10.1016/j.jsc.2023.102263","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102263","url":null,"abstract":"<div><p>The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49755857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonathan D. Hauenstein, Yang-Hui He, Ilias Kotsireas, Dhagash Mehta, Tingting Tang
{"title":"Special issue on Algebraic Geometry and Machine Learning","authors":"Jonathan D. Hauenstein, Yang-Hui He, Ilias Kotsireas, Dhagash Mehta, Tingting Tang","doi":"10.1016/j.jsc.2022.10.003","DOIUrl":"https://doi.org/10.1016/j.jsc.2022.10.003","url":null,"abstract":"","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Segre-driven radicality testing","authors":"Martin Helmer , Elias Tsigaridas","doi":"10.1016/j.jsc.2023.102262","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102262","url":null,"abstract":"<div><p><span><span>We present a probabilistic algorithm to test if a </span>homogeneous polynomial ideal </span><em>I</em> defining a scheme <em>X</em> in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is radical using Segre classes and other geometric notions from intersection theory which is applicable for certain classes of ideals. If all isolated primary components of the scheme <em>X</em> are reduced and it has no embedded components outside of the singular locus of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>red</mi></mrow></msub><mo>=</mo><mi>V</mi><mo>(</mo><msqrt><mrow><mi>I</mi></mrow></msqrt><mo>)</mo></math></span>, then the algorithm is not applicable and will return that it is unable to decide radically; in all the other cases it will terminate successfully and in either case its complexity is singly exponential in <em>n</em><span><span>. The realm of the ideals for which our radical testing procedure is applicable and for which it requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly </span>exponential complexity for the ideal membership problem.</span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Invariants of SDP exactness in quadratic programming","authors":"Julia Lindberg , Jose Israel Rodriguez","doi":"10.1016/j.jsc.2023.102258","DOIUrl":"10.1016/j.jsc.2023.102258","url":null,"abstract":"<div><p><span>In this paper we study the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. We first give conditions under which this region is invariant under the choice of generators defining the feasible set. We then describe this region when the feasible set is invariant under the action of a subgroup of the </span>general linear group. We conclude by applying these results to quadratic binary programs. We give an explicit description of objective functions where the Shor relaxation is exact and use this knowledge to design an algorithm that produces candidate solutions for binary quadratic programs.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49349822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four","authors":"Nancy Abdallah , Hal Schenck","doi":"10.1016/j.jsc.2023.102257","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102257","url":null,"abstract":"<div><p>In (<span>Stanley, 1978</span>), Stanley constructs an example of an Artinian Gorenstein (AG) ring <em>A</em> with non-unimodal <em>H</em>-vector <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. Migliore-Zanello show in (<span>Migliore and Zanello, 2017</span>) that for regularity <span><math><mi>r</mi><mo>=</mo><mn>4</mn></math></span><span>, Stanley's example has the smallest possible codimension </span><em>c</em> for an AG ring with non-unimodal <em>H</em>-vector.</p><p>The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal <em>H</em>-vector fails to have WLP. In codimension <span><math><mi>c</mi><mo>=</mo><mn>3</mn></math></span> it is conjectured that all AG rings have WLP. For <span><math><mi>c</mi><mo>=</mo><mn>4</mn></math></span>, Gondim shows in (<span>Gondim, 2017</span>) that WLP always holds for <span><math><mi>r</mi><mo>≤</mo><mn>4</mn></math></span> and gives a family where WLP fails for any <span><math><mi>r</mi><mo>≥</mo><mn>7</mn></math></span>, building on Ikeda's example (<span>Ikeda, 1996</span>) of failure for <span><math><mi>r</mi><mo>=</mo><mn>5</mn></math></span>. In this note we study the minimal free resolution of <em>A</em> and relation to Lefschetz properties (both weak and strong) and Jordan type for <span><math><mi>c</mi><mo>=</mo><mn>4</mn></math></span> and <span><math><mi>r</mi><mo>≤</mo><mn>6</mn></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Algebraic number fields and the LLL algorithm","authors":"M.J. Uray","doi":"10.1016/j.jsc.2023.102261","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102261","url":null,"abstract":"<div><p><span>In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let </span><em>K</em> be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in <em>K</em> in terms of the size of the input and the parameters of <em>K</em>. We include some earlier results about these, but we go further than them, e.g. we also analyze some <span><math><mi>R</mi></math></span>-specific operations in <em>K</em> like less-than comparison.</p><p><span>In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from </span><span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and give a polynomial upper bound on the running time when the computations in <em>K</em> are performed exactly (as opposed to floating-point approximations).</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The critical curvature degree of an algebraic variety","authors":"Emil Horobeţ","doi":"10.1016/j.jsc.2023.102259","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102259","url":null,"abstract":"<div><p>In this article we study the complexity involved in the computation of the reach in arbitrary dimension and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing them.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Universal equations for maximal isotropic Grassmannians","authors":"Tim Seynnaeve, Nafie Tairi","doi":"10.1016/j.jsc.2023.102260","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102260","url":null,"abstract":"<div><p>The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of <span><math><mi>G</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>iso</mi></mrow></msub><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></span> or <span><math><mi>G</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>iso</mi></mrow></msub><mo>(</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero","authors":"Jiayue Qi","doi":"10.1016/j.jsc.2023.102253","DOIUrl":"10.1016/j.jsc.2023.102253","url":null,"abstract":"<div><p><span>The Chow ring of the moduli space of marked rational curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we provide an algorithm for computing the intersection degree of tuples of Keel's divisor classes — we call it the forest algorithm; the complexity of which is </span><span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> in the worst case, where <em>n</em> refers to the number of marks in the ambient moduli space. Last but not least, we introduce three identities on multinomial coefficients which naturally came into play, showing that they are all equivalent to the correctness of the base case of the forest algorithm.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49013075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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