Journal of Symbolic Computation最新文献

{"title":"Short proofs of ideal membership","authors":"Clemens Hofstadler ,&nbsp;Thibaut Verron","doi":"10.1016/j.jsc.2024.102325","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102325","url":null,"abstract":"<div><p>A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting.</p><p>We show that the problem of computing cofactor representations with a bounded number of terms is decidable and <span><math><mtext>NP</mtext></math></span>-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gröbner basis algorithms. We show that, for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000294/pdfft?md5=fc06471a76a7e331737ea355a494162b&pid=1-s2.0-S0747717124000294-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540319","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":"On the connectedness of multistationarity regions of small reaction networks","authors":"Allison McClure,&nbsp;Anne Shiu","doi":"10.1016/j.jsc.2024.102323","DOIUrl":"10.1016/j.jsc.2024.102323","url":null,"abstract":"<div><p>A multistationarity region is the part of a reaction network's parameter space that gives rise to multiple steady states. Mathematically, this region consists of the positive parameters for which a parametrized family of polynomial equations admits two or more positive roots. Much recent work has focused on analyzing multistationarity regions of biologically significant reaction networks and determining whether such regions are connected; indeed, a better understanding of the topology and geometry of such regions may help elucidate how robust multistationarity is to perturbations. Here we focus on the multistationarity regions of small networks, those with few species and few reactions. For two families of such networks – those with one species and up to three reactions, and those with two species and up to two reactions – we prove that the resulting multistationarity regions are connected. We also give an example of a network with one species and six reactions for which the multistationarity region is disconnected. Our proofs rely on the formula for the discriminant of a trinomial, a classification of small multistationary networks, and a recent result of Feliu and Telek that partially generalizes Descartes' rule of signs.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196310","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 integral closure of a primary ideal is not always primary","authors":"Nan Li ,&nbsp;Zijia Li ,&nbsp;Zhi-Hong Yang ,&nbsp;Lihong Zhi","doi":"10.1016/j.jsc.2024.102315","DOIUrl":"10.1016/j.jsc.2024.102315","url":null,"abstract":"<div><p>In <span>1936</span>, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>. We provide counterexamples to Krull's question regarding polynomial rings over any fields. We also find that the Jacobian ideal <em>J</em> of the polynomial <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>z</mi><mi>t</mi><mo>+</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> given by <span>Briançon and Speder (1975)</span> is a counterexample to Krull's question.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071563","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":"Solving polynomial systems over non-fields and applications to modular polynomial factoring","authors":"Sayak Chakrabarti,&nbsp;Ashish Dwivedi,&nbsp;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>&gt;</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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}