Correctness of results from mixed-integer linear programming (MILP) solvers is critical, particularly in the context of applications such as hardware verification, compiler optimization, or machine-assisted theorem proving. To this end, VIPR 1.0 is the first recently proposed general certificate format for answers produced by MILP solvers. We design a schema to encode VIPR's inference rules as a ground formula that completely characterizes the validity of the algorithmic check, removing any ambiguities and imprecisions present in the specification. We formally verify the correctness of our schema at the logical level using Why3's automated deductive logic framework. Furthermore, we implement a checker for VIPR certificates by expressing our formally verified ground formula with the Satisfiability Modulo Theory Library (SMT-LIB) and check its validity. Our approach is solver-agnostic, and we test its viability using benchmark instances found in the literature.
{"title":"Satisfiability modulo theories for verifying MILP certificates","authors":"Kenan Wood , Runtian Zhou , Haoze Wu , Hammurabi Mendes , Jonad Pulaj","doi":"10.1016/j.jsc.2025.102543","DOIUrl":"10.1016/j.jsc.2025.102543","url":null,"abstract":"<div><div>Correctness of results from mixed-integer linear programming (MILP) solvers is critical, particularly in the context of applications such as hardware verification, compiler optimization, or machine-assisted theorem proving. To this end, VIPR 1.0 is the first recently proposed general certificate format for answers produced by MILP solvers. We design a schema to encode VIPR's inference rules as a ground formula that completely characterizes the validity of the algorithmic check, removing any ambiguities and imprecisions present in the specification. We formally verify the correctness of our schema at the logical level using <span>Why3</span>'s automated deductive logic framework. Furthermore, we implement a checker for VIPR certificates by expressing our formally verified ground formula with the Satisfiability Modulo Theory Library (SMT-LIB) and check its validity. Our approach is solver-agnostic, and we test its viability using benchmark instances found in the literature.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102543"},"PeriodicalIF":1.1,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924765","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}
Pub Date : 2025-12-23DOI: 10.1016/j.jsc.2025.102546
Matěj Doležálek , Nikhil Ken
We prove that for any , , all secant varieties of the Segre-Veronese variety have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the inductant. With a combinatorial investigation of these constructions, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.
{"title":"Secant varieties of Segre-Veronese varieties Pm×Pn embedded by O(1,2) are non-defective for n ≫ m3, m ≥ 3","authors":"Matěj Doležálek , Nikhil Ken","doi":"10.1016/j.jsc.2025.102546","DOIUrl":"10.1016/j.jsc.2025.102546","url":null,"abstract":"<div><div>We prove that for any <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>n</mi><mo>≫</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, all secant varieties of the Segre-Veronese variety <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the <em>inductant</em>. With a combinatorial investigation of these constructions, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102546"},"PeriodicalIF":1.1,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924777","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}
Pub Date : 2025-12-22DOI: 10.1016/j.jsc.2025.102545
Guo-Shuai Mao
In this paper, we mainly prove a challenging congruence conjecture of Z.-W. Sun (2014) via symbolic summation: Let be a prime. Then where denotes the n-th harmonic number. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities.
{"title":"Supercongruences involving products of two binomial coefficients modulo p4","authors":"Guo-Shuai Mao","doi":"10.1016/j.jsc.2025.102545","DOIUrl":"10.1016/j.jsc.2025.102545","url":null,"abstract":"<div><div>In this paper, we mainly prove a challenging congruence conjecture of Z.-W. Sun (<span><span>2014</span></span>) via symbolic summation: Let <span><math><mi>p</mi><mo>></mo><mn>5</mn></math></span> be a prime. Then<span><span><span><math><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>k</mi><msup><mrow><mn>16</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mo>≡</mo><mo>−</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>H</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denotes the <em>n</em>-th harmonic number. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102545"},"PeriodicalIF":1.1,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839832","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}
Pub Date : 2025-12-22DOI: 10.1016/j.jsc.2025.102544
Ya-Lun Tsai
In a single species population model on n identical patches with strong Allee effect and spatial dispersal, we study the steady states of the system with respect to the same Allee threshold β for all patches and the same dispersal rate α if they are connected. In the cases of connected patches with complete digraphs, we find the bifurcation polynomials for all through Jacobian matrices, and classify the numbers of steady states completely for n up to 14. For any fixed n, if , there are only three steady states, two stables and one unstable that generalizes the case of . All the bifurcation polynomials are obtained efficiently by computing Gröbner bases with elimination orderings for systems containing Jacobian determinants.
On the other hand, for , we count the numbers of steady states for all possible connectivities through considering all non-isomorphic digraphs with three nodes. For most connectivities, the numbers of steady states do not always decrease monotonously when β is fixed and α increases as the previous known cases. Interestingly, there are some connectivities where the number of steady states does not eventually reduce to the minimal. Here, if the Gröbner basis computation is not feasible with Jacobian determinants to obtain the bifurcation polynomials, we present a method to compute them by simply resultants for two polynomials to eliminate variables without using Jacobian determinants of order greater than one.
{"title":"Bifurcations in a population model on N patches with strong Allee effect and spatial dispersal through Jacobian matrices","authors":"Ya-Lun Tsai","doi":"10.1016/j.jsc.2025.102544","DOIUrl":"10.1016/j.jsc.2025.102544","url":null,"abstract":"<div><div>In a single species population model on <em>n</em> identical patches with strong Allee effect and spatial dispersal, we study the steady states of the system with respect to the same Allee threshold <em>β</em> for all patches and the same dispersal rate <em>α</em> if they are connected. In the cases of connected patches with complete digraphs, we find the bifurcation polynomials for all <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> through Jacobian matrices, and classify the numbers of steady states completely for <em>n</em> up to 14. For any fixed <em>n</em>, if <span><math><mi>α</mi><mo>≥</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, there are only three steady states, two stables and one unstable that generalizes the case of <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>. All the bifurcation polynomials are obtained efficiently by computing Gröbner bases with elimination orderings for systems containing Jacobian determinants.</div><div>On the other hand, for <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>, we count the numbers of steady states for all possible connectivities through considering all non-isomorphic digraphs with three nodes. For most connectivities, the numbers of steady states do not always decrease monotonously when <em>β</em> is fixed and <em>α</em> increases as the previous known cases. Interestingly, there are some connectivities where the number of steady states does not eventually reduce to the minimal. Here, if the Gröbner basis computation is not feasible with Jacobian determinants to obtain the bifurcation polynomials, we present a method to compute them by simply resultants for two polynomials to eliminate variables without using Jacobian determinants of order greater than one.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102544"},"PeriodicalIF":1.1,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883878","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}
Pub Date : 2025-12-10DOI: 10.1016/j.jsc.2025.102542
Hao Liang, Jingyu Lu, Manolis C. Tsakiris, Lihong Zhi
Unlabeled sensing is the problem of solving a linear system of equations, where the right-hand-side vector is known only up to a permutation. In this work, we study fields of rational functions related to symmetric polynomials and their images under a linear projection of the variables; as a consequence, we establish that the solution to an n-dimensional unlabeled sensing problem with generic data can be obtained as the unique solution to a system of polynomial equations of degrees in n unknowns. Besides the new theoretical insights, this development offers the potential for scaling up algebraic unlabeled sensing algorithms.
{"title":"A field-theoretic view of unlabeled sensing","authors":"Hao Liang, Jingyu Lu, Manolis C. Tsakiris, Lihong Zhi","doi":"10.1016/j.jsc.2025.102542","DOIUrl":"10.1016/j.jsc.2025.102542","url":null,"abstract":"<div><div>Unlabeled sensing is the problem of solving a linear system of equations, where the right-hand-side vector is known only up to a permutation. In this work, we study fields of rational functions related to symmetric polynomials and their images under a linear projection of the variables; as a consequence, we establish that the solution to an <em>n</em>-dimensional unlabeled sensing problem with generic data can be obtained as the unique solution to a system of <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> polynomial equations of degrees <span><math><mn>1</mn><mo>,</mo><mspace></mspace><mn>2</mn><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span> in <em>n</em> unknowns. Besides the new theoretical insights, this development offers the potential for scaling up algebraic unlabeled sensing algorithms.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102542"},"PeriodicalIF":1.1,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839833","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}
Pub Date : 2025-12-04DOI: 10.1016/j.jsc.2025.102541
Alvaro Gonzalez-Hernandez
We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the p-rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.
{"title":"Explicit desingularisation of Kummer surfaces in characteristic two via specialisation","authors":"Alvaro Gonzalez-Hernandez","doi":"10.1016/j.jsc.2025.102541","DOIUrl":"10.1016/j.jsc.2025.102541","url":null,"abstract":"<div><div>We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the <em>p</em>-rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102541"},"PeriodicalIF":1.1,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839834","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}
Pub Date : 2025-12-03DOI: 10.1016/j.jsc.2025.102540
Anthony Pisani
We employ the recently developed hybrid and mmgroup computational models for groups to calculate the character table of , a maximal subgroup of the Monster sporadic simple group. This completes the list of the character tables of maximal subgroups of the Monster. Our approach illustrates how the aforementioned computational models can be used to calculate relatively straightforwardly in the Monster.
{"title":"Computing the character table of a 2-local maximal subgroup of the Monster","authors":"Anthony Pisani","doi":"10.1016/j.jsc.2025.102540","DOIUrl":"10.1016/j.jsc.2025.102540","url":null,"abstract":"<div><div>We employ the recently developed hybrid and <span>mmgroup</span> computational models for groups to calculate the character table of <figure><img></figure>, a maximal subgroup of the Monster sporadic simple group. This completes the list of the character tables of maximal subgroups of the Monster. Our approach illustrates how the aforementioned computational models can be used to calculate relatively straightforwardly in the Monster.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102540"},"PeriodicalIF":1.1,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737432","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}
Pub Date : 2025-11-24DOI: 10.1016/j.jsc.2025.102533
Lilybelle Cowland Kellock, Elisa Lorenzo García
Tate's algorithm tells us that for an elliptic curve E over a local field K of residue characteristic ≥5, has potentially good reduction if and only if . It also tells us that when is semistable the dual graph of the special fibre of the minimal regular model of can be recovered from . We generalise these results to hyperelliptic curves of genus over local fields of odd residue characteristic K by defining a list of absolute invariants that determine the potential stable model of a genus g hyperelliptic curve C. They also determine the dual graph of the special fibre of the minimal regular model of if is semistable. This list depends only on the genus of C, and the absolute invariants can be written in terms of the coefficients of a Weierstrass equation for C. We explicitly describe the method by which the valuations of the invariants recover the dual graphs. Additionally, we show by way of a counterexample that if , there is no list of invariants whose valuations determine the dual graph of the special fibre of the minimal regular model of a genus g hyperelliptic curve C over a local field K of odd residue characteristic when C is not assumed to be semistable.
{"title":"Invariants recovering the reduction type of a hyperelliptic curve","authors":"Lilybelle Cowland Kellock, Elisa Lorenzo García","doi":"10.1016/j.jsc.2025.102533","DOIUrl":"10.1016/j.jsc.2025.102533","url":null,"abstract":"<div><div>Tate's algorithm tells us that for an elliptic curve <em>E</em> over a local field <em>K</em> of residue characteristic ≥5, <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> has potentially good reduction if and only if <span><math><mtext>ord</mtext><mo>(</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>)</mo><mo>≥</mo><mn>0</mn></math></span>. It also tells us that when <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> is semistable the dual graph of the special fibre of the minimal regular model of <span><math><mi>E</mi><mo>/</mo><msup><mrow><mi>K</mi></mrow><mrow><mtext>unr</mtext></mrow></msup></math></span> can be recovered from <span><math><mtext>ord</mtext><mo>(</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>)</mo></math></span>. We generalise these results to hyperelliptic curves of genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span> over local fields of odd residue characteristic <em>K</em> by defining a list of absolute invariants that determine the potential stable model of a genus <em>g</em> hyperelliptic curve <em>C</em>. They also determine the dual graph of the special fibre of the minimal regular model of <span><math><mi>C</mi><mo>/</mo><msup><mrow><mi>K</mi></mrow><mrow><mtext>unr</mtext></mrow></msup></math></span> if <span><math><mi>C</mi><mo>/</mo><mi>K</mi></math></span> is semistable. This list depends only on the genus of <em>C</em>, and the absolute invariants can be written in terms of the coefficients of a Weierstrass equation for <em>C</em>. We explicitly describe the method by which the valuations of the invariants recover the dual graphs. Additionally, we show by way of a counterexample that if <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span>, there is no list of invariants whose valuations determine the dual graph of the special fibre of the minimal regular model of a genus <em>g</em> hyperelliptic curve <em>C</em> over a local field <em>K</em> of odd residue characteristic when <em>C</em> is not assumed to be semistable.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102533"},"PeriodicalIF":1.1,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625258","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}
Pub Date : 2025-11-21DOI: 10.1016/j.jsc.2025.102532
Rémi Prébet , Mohab Safey El Din , Éric Schost
A roadmap for an algebraic set V defined by polynomials with coefficients in the field of rational numbers is an algebraic curve contained in V whose intersection with all connected components of is connected. These objects, introduced by Canny, can be used to answer connectivity queries over provided that they are required to contain the finite set of query points ; in this case, we say that the roadmap is associated to .
In this paper, we make effective a connectivity result we previously proved, to design a Monte Carlo algorithm which, on input (i) a finite sequence of polynomials defining V (and satisfying some regularity assumptions) and (ii) an algebraic representation of finitely many query points in V, computes a roadmap for . This algorithm generalizes the nearly optimal one introduced by the last two authors by dropping a boundedness assumption on the real trace of V.
The output size and running times of our algorithm are both polynomial in , where D is the maximal degree of the input equations and d is the dimension of V. As far as we know, the best previously known algorithm dealing with such sets has an output size and running time respectively polynomial in and .
{"title":"Computing roadmaps in unbounded smooth real algebraic sets II: Algorithm and complexity","authors":"Rémi Prébet , Mohab Safey El Din , Éric Schost","doi":"10.1016/j.jsc.2025.102532","DOIUrl":"10.1016/j.jsc.2025.102532","url":null,"abstract":"<div><div>A roadmap for an algebraic set <em>V</em> defined by polynomials with coefficients in the field <span><math><mi>Q</mi></math></span> of rational numbers is an algebraic curve contained in <em>V</em> whose intersection with all connected components of <span><math><mi>V</mi><mo>∩</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is connected. These objects, introduced by Canny, can be used to answer connectivity queries over <span><math><mi>V</mi><mo>∩</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> provided that they are required to contain the finite set of query points <span><math><mi>P</mi><mo>⊂</mo><mi>V</mi></math></span>; in this case, we say that the roadmap is associated to <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>.</div><div>In this paper, we make effective a connectivity result we previously proved, to design a Monte Carlo algorithm which, on input <em>(i)</em> a finite sequence of polynomials defining <em>V</em> (and satisfying some regularity assumptions) and <em>(ii)</em> an algebraic representation of finitely many query points <span><math><mi>P</mi></math></span> in <em>V</em>, computes a roadmap for <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>. This algorithm generalizes the nearly optimal one introduced by the last two authors by dropping a boundedness assumption on the real trace of <em>V</em>.</div><div>The output size and running times of our algorithm are both polynomial in <span><math><msup><mrow><mo>(</mo><mi>n</mi><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi><mi>log</mi><mo></mo><mi>d</mi></mrow></msup></math></span>, where <em>D</em> is the maximal degree of the input equations and <em>d</em> is the dimension of <em>V</em>. As far as we know, the best previously known algorithm dealing with such sets has an output size and running time respectively polynomial in <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></msup><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi></mrow></msup></math></span> and <span><math><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></msup><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102532"},"PeriodicalIF":1.1,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685319","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jsc.2025.102529
Gabriel Mattos Langeloh
Families of integer programming problems can be solved efficiently in practice once their reduced Gröbner basis is known. However, computing Gröbner bases is often hard, especially when binary or integer bounded variables are present in the problem formulation. In this paper, we study the specific structure of the constraint matrix of integer programs with bounded variables and the implications of this structure to the truncated Gröbner bases of integer programming problems. In this direction, we introduce a new Binary Truncation Criterion that is capable of predicting and eliminating useless S-vectors before they built in the Gröbner basis computation. Additionally, we propose improvements to the Gröbner basis approach to multiobjective integer programming of Jiménez-Tafur (2017) and Hartillo-Hermoso et al. (2020), such as a proof that truncated Gröbner bases can be used in their algorithm with no loss of correctness, implying that our new truncation techniques are also useful in this application. All new proposed methods are implemented in the open source package IPGBs and their performance is empirically validated.
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