Journal of Symbolic Computation最新文献

{"title":"Solving polynomial systems over non-fields and applications to modular polynomial factoring","authors":"Sayak Chakrabarti, Ashish Dwivedi, Nitin Saxena","doi":"10.1016/j.jsc.2024.102314","DOIUrl":"10.1016/j.jsc.2024.102314","url":null,"abstract":"<div><p>We study the problem of solving a system of <em>m</em> polynomials in <em>n</em> variables over the ring of integers modulo a prime-power <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>. The problem over finite fields is well studied in varied parameter settings. For small characteristic <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, Lokshtanov et al. (SODA'17) initiated the study, for degree <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> systems, to improve the exhaustive search complexity of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>⋅</mo><mtext>poly</mtext><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>0.8765</mn><mi>n</mi></mrow></msup><mo>)</mo><mo>⋅</mo><mtext>poly</mtext><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>; which currently is improved to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>0.6943</mn><mi>n</mi></mrow></msup><mo>)</mo><mo>⋅</mo><mtext>poly</mtext><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> in Dinur (SODA'21). For large <em>p</em> but constant <em>n</em>, Huang and Wong (FOCS'96) gave a randomized <span><math><mtext>poly</mtext><mo>(</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></math></span> time algorithm. Note that for growing <em>n</em>, system-solving is known to be <em>intractable</em> even with <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and degree <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>.</p><p>We devise a randomized <span><math><mtext>poly</mtext><mo>(</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></math></span>-time algorithm to find a root of a given system of <em>m</em> integral polynomials of degrees bounded by <em>d</em>, in <em>n</em> variables, modulo a prime power <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>; when <span><math><mi>n</mi><mo>+</mo><mi>k</mi></math></span> is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic <em>p</em>) to system-solving over Galois <em>rings</em> (i.e., characteristic <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>); when <span><math><mi>k</mi><mo>></mo><mn>1</mn></math></span> is constant. The challenge here is to find a lift of <em>singular</em> <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.</p><p>Our algorithm has applications to factoring univariate polynomials over Galois rings. Given <span><math><mi>f</mi><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102314"},"PeriodicalIF":0.7,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071367","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 computational approach to almost-inner derivations","authors":"Heiko Dietrich ,&nbsp;Willem A. de Graaf","doi":"10.1016/j.jsc.2024.102312","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102312","url":null,"abstract":"<div><p>We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102312"},"PeriodicalIF":0.7,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000166/pdfft?md5=158ca0ceced5645dd3d6b5c19e5bfa5f&pid=1-s2.0-S0747717124000166-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140051681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stabilized recovery and model reduction for multivariate exponential polynomials","authors":"Juan Manuel Peña ,&nbsp;Tomas Sauer","doi":"10.1016/j.jsc.2024.102313","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102313","url":null,"abstract":"<div><p>Recovery of multivariate exponential polynomials, i.e., the multivariate version of Prony's problem, can be stabilized by using more than the minimally needed multiinteger samples of the function. We present an algorithm that takes into account this extra information and prove a backward error estimate for the algebraic recovery method SMILE. In addition, we give a method to approximate data by an exponential polynomial sequence of a given structure as a step in the direction of multivariate model reduction.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102313"},"PeriodicalIF":0.7,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000178/pdfft?md5=a03807abfc64e6721e202a9e27a5dbdf&pid=1-s2.0-S0747717124000178-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140051680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing a group action from the class field theory of imaginary hyperelliptic function fields","authors":"Antoine Leudière,&nbsp;Pierre-Jean Spaenlehauer","doi":"10.1016/j.jsc.2024.102311","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102311","url":null,"abstract":"<div><p>We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed <em>τ</em>-degree between Drinfeld <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span>-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102311"},"PeriodicalIF":0.7,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140051678","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":"Matrix factorizations of the discriminant of Sn","authors":"Eleonore Faber ,&nbsp;Colin Ingalls ,&nbsp;Simon May ,&nbsp;Marco Talarico","doi":"10.1016/j.jsc.2024.102310","DOIUrl":"10.1016/j.jsc.2024.102310","url":null,"abstract":"<div><p>Consider the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> acting as a reflection group on the polynomial ring <span><math><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> where <em>k</em> is a field, such that Char(<em>k</em>) does not divide <em>n</em>!. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of <em>n</em> and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen–Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and indicate how to generalize them to the reflection groups <span><math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102310"},"PeriodicalIF":0.7,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000142/pdfft?md5=688baaec9b10f27e6369b86c65d8e101&pid=1-s2.0-S0747717124000142-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved complexity bound for computing the topology of a real algebraic space curve","authors":"Jin-San Cheng ,&nbsp;Kai Jin ,&nbsp;Marc Pouget ,&nbsp;Junyi Wen ,&nbsp;Bingwei Zhang","doi":"10.1016/j.jsc.2024.102309","DOIUrl":"10.1016/j.jsc.2024.102309","url":null,"abstract":"<div><p>We propose a new algorithm to compute the topology of a real algebraic space curve. The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weaker notion of generic position. As distinct to previous work, our <em>sweep generic position</em> does not require that <em>x</em>-critical points have different <em>x</em>-coordinates. The complexity of achieving this sweep generic position property is thus no longer a bottleneck in term of complexity. The bit complexity of our algorithm is <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>18</mn></mrow></msup><mo>+</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>17</mn></mrow></msup><mi>τ</mi><mo>)</mo></math></span> where <em>d</em> and <em>τ</em> bound the degree and the bitsize of the integer coefficients, respectively, of the defining polynomials of the curve and polylogarithmic factors are ignored. To the best of our knowledge, this improves upon the best currently known results at least by a factor of <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102309"},"PeriodicalIF":0.7,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919633","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":"Rational solutions to the first order difference equations in the bivariate difference field","authors":"Qing-Hu Hou ,&nbsp;Yarong Wei","doi":"10.1016/j.jsc.2024.102308","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102308","url":null,"abstract":"<div><p>Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form <span><math><mi>a</mi><mi>σ</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>+</mo><mi>b</mi><mi>g</mi><mo>=</mo><mi>f</mi></math></span> in the bivariate difference field <span><math><mo>(</mo><mi>F</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>f</mi><mo>∈</mo><mi>F</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> are known binary functions of <em>α</em>, <em>β</em>, and <em>α</em>, <em>β</em> are two algebraically independent transcendental elements, <em>σ</em> is a transformation that satisfies <span><math><mi>σ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>=</mo><mi>β</mi></math></span>, <span><math><mi>σ</mi><mo>(</mo><mi>β</mi><mo>)</mo><mo>=</mo><mi>u</mi><mi>α</mi><mo>+</mo><mi>v</mi><mi>β</mi></math></span>, where <span><math><mi>u</mi><mo>,</mo><mi>v</mi><mo>≠</mo><mn>0</mn><mo>∈</mo><mi>F</mi></math></span>. Based on it, we then describe algorithms for finding the universal denominator for those equations in the bivariate difference field under certain assumptions. This reduces the general problem of finding the rational solutions of such equations to the problem of finding the polynomial solutions of such equations.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"124 ","pages":"Article 102308"},"PeriodicalIF":0.7,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139901497","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":"Tensor decompositions on simplicial complexes with invariance","authors":"Gemma De las Cuevas ,&nbsp;Matt Hoogsteder Riera ,&nbsp;Tim Netzer","doi":"10.1016/j.jsc.2024.102299","DOIUrl":"10.1016/j.jsc.2024.102299","url":null,"abstract":"<div><p><span>Tensors are ubiquitous in mathematics and the sciences, as they allow to store information in a concise way. Decompositions of tensors may give insights into their structure and complexity. In this work, we develop a new framework for decompositions of tensors, taking into account invariance, positivity and a geometric arrangement of their local spaces. We define an invariant decomposition with indices arranged on a simplicial complex which is explicitly invariant under a group action. We give a constructive proof that this decomposition exists for all invariant tensors, after possibly enriching the simplicial complex. We further define several decompositions certifying positivity, and prove similar existence results, as well as inequalities between the corresponding ranks. Our results generalize results from the theory of </span>tensor networks, used in the study of quantum many-body systems, for example.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"124 ","pages":"Article 102299"},"PeriodicalIF":0.7,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139497517","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":"Computing the binomial part of a polynomial ideal","authors":"Martin Kreuzer,&nbsp;Florian Walsh","doi":"10.1016/j.jsc.2024.102298","DOIUrl":"10.1016/j.jsc.2024.102298","url":null,"abstract":"<div><p>Given an ideal <em>I</em> in a polynomial ring <span><math><mi>K</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> over a field <em>K</em>, we present a complete algorithm to compute the binomial part of <em>I</em>, i.e., the subideal <span><math><mrow><mi>Bin</mi></mrow><mo>(</mo><mi>I</mi><mo>)</mo></math></span> of <em>I</em> generated by all monomials and binomials in <em>I</em>. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional <em>K</em>-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic <em>p</em>. Next we examine the computation of unit lattices in finitely generated <em>K</em>-algebras, as well as their associated characters and lattice ideals. This allows us to calculate <span><math><mrow><mi>Bin</mi></mrow><mo>(</mo><mi>I</mi><mo>)</mo></math></span> when <em>I</em> is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of <span><math><mrow><mi>Bin</mi></mrow><mo>(</mo><mi>I</mi><mo>)</mo></math></span> for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called <span><math><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-binomial parts. All algorithms have been implemented in <span>SageMath</span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"124 ","pages":"Article 102298"},"PeriodicalIF":0.7,"publicationDate":"2024-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000026/pdfft?md5=bc32bb62dcb12f7f2c1d113994ec49bf&pid=1-s2.0-S0747717124000026-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A post-quantum key exchange protocol from the intersection of conics","authors":"Alberto Alzati, Daniele Di Tullio, Manoj Gyawali, Alfonso Tortora","doi":"10.1016/j.jsc.2024.102297","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102297","url":null,"abstract":"<p>In this paper we present a key exchange protocol in which Alice and Bob have secret keys given by two conics embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded curves. Both of them construct some common invariants given by the intersection of two conics.</p>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102670","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}