Pub Date : 2025-04-15DOI: 10.1016/j.jsc.2025.102450
Elizabeth Pratt , Bernd Sturmfels
The classical Chow form encodes any projective variety by one equation. We here introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas. These were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra, and we develop his approach further. Universal formulas for branch loci are obtained from Hurwitz-Lam forms. Our focus is on computations and applications in geometry.
{"title":"The Chow-Lam form","authors":"Elizabeth Pratt , Bernd Sturmfels","doi":"10.1016/j.jsc.2025.102450","DOIUrl":"10.1016/j.jsc.2025.102450","url":null,"abstract":"<div><div>The classical Chow form encodes any projective variety by one equation. We here introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas. These were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra, and we develop his approach further. Universal formulas for branch loci are obtained from Hurwitz-Lam forms. Our focus is on computations and applications in geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102450"},"PeriodicalIF":0.6,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869618","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 : 2025-04-14DOI: 10.1016/j.jsc.2025.102448
Gianni Petrella
We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with QuiverTools, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.
{"title":"Partial semiorthogonal decompositions for quiver moduli","authors":"Gianni Petrella","doi":"10.1016/j.jsc.2025.102448","DOIUrl":"10.1016/j.jsc.2025.102448","url":null,"abstract":"<div><div>We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with <span>QuiverTools</span>, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102448"},"PeriodicalIF":0.6,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839767","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}
Computing the arrangement of quadrics in 3D is a fundamental problem in symbolic computation, with challenges arising when handling degenerate cases and asymptotic critical values. State-of-the-art methods typically require a generic change of coordinates to manage these asymptotes, rendering certain problems intractable. A specific instance of this challenge appears in digital geometry, where comparing 3D shapes up to isometry requires applying a 3D rigid motion on and mapping the result back to , a process typically achieved via a digitization operator. However, such motions do not preserve the topology of digital objects, making the analysis of digitized rigid motions crucial. Our main contribution is the decomposition of the 6D parameter space of digitized rigid motions for image patches of radius up to three. This problem reduces to computing the arrangement of up to 741 quadrics, some of which are degenerate. To address the computational challenges, we introduce and implement a new algorithm for computing arrangements of quadrics in 3D, specifically designed to handle degenerate directions and asymptotic critical values. This approach allows us to overcome the limitations of existing methods, making the problem tractable in the context of digital geometry.
{"title":"On arrangements of quadrics in decomposing the parameter space of 3D digitized rigid motions","authors":"Kacper Pluta , Guillaume Moroz , Yukiko Kenmochi , Pascal Romon","doi":"10.1016/j.jsc.2025.102447","DOIUrl":"10.1016/j.jsc.2025.102447","url":null,"abstract":"<div><div>Computing the arrangement of quadrics in 3D is a fundamental problem in symbolic computation, with challenges arising when handling degenerate cases and asymptotic critical values. State-of-the-art methods typically require a generic change of coordinates to manage these asymptotes, rendering certain problems intractable. A specific instance of this challenge appears in digital geometry, where comparing 3D shapes up to isometry requires applying a 3D rigid motion on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and mapping the result back to <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, a process typically achieved via a digitization operator. However, such motions do not preserve the topology of digital objects, making the analysis of digitized rigid motions crucial. Our main contribution is the decomposition of the 6D parameter space of digitized rigid motions for image patches of radius up to three. This problem reduces to computing the arrangement of up to 741 quadrics, some of which are degenerate. To address the computational challenges, we introduce and implement a new algorithm for computing arrangements of quadrics in 3D, specifically designed to handle degenerate directions and asymptotic critical values. This approach allows us to overcome the limitations of existing methods, making the problem tractable in the context of digital geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102447"},"PeriodicalIF":0.6,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143855157","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 : 2025-04-02DOI: 10.1016/j.jsc.2025.102446
Erin Connelly , Felix Rydell
In recent work, algebraic computational software was used to provide the exact algebraic conditions under which a six-tuple of fundamental matrices, corresponding to 4 images, is compatible, i.e., there exist 4 cameras such that each pair has the appropriate fundamental matrix. It has been further demonstrated that quadruplewise compatibility is sufficient when the number of cameras greater than 4. We expand on these prior results by proving equivalent geometric conditions for compatibility. We find that compatibility can be characterized via the intersections of epipolar lines in one of the images.
{"title":"Geometric interpretations of compatibility for fundamental matrices","authors":"Erin Connelly , Felix Rydell","doi":"10.1016/j.jsc.2025.102446","DOIUrl":"10.1016/j.jsc.2025.102446","url":null,"abstract":"<div><div>In recent work, algebraic computational software was used to provide the exact algebraic conditions under which a six-tuple of fundamental matrices, corresponding to 4 images, is compatible, i.e., there exist 4 cameras such that each pair has the appropriate fundamental matrix. It has been further demonstrated that quadruplewise compatibility is sufficient when the number of cameras greater than 4. We expand on these prior results by proving equivalent geometric conditions for compatibility. We find that compatibility can be characterized via the intersections of epipolar lines in one of the images.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102446"},"PeriodicalIF":0.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817118","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 : 2025-04-02DOI: 10.1016/j.jsc.2025.102445
Martin Kreuzer , Lorenzo Robbiano
Let K be a field and . The technique of elimination by substitution is based on discovering a coherently -separating tuple of polynomials in an ideal I, i.e., on finding polynomials such that with . Here we elaborate on this technique in the case when P is non-negatively graded. The existence of a coherently Z-separating tuple is reduced to solving several -module membership problems. Best separable re-embeddings, i.e., isomorphisms with maximal #Z, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing as a fibration over an affine space, we show that its fibers allow optimal Z-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of P such that additional Z-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between and a non-negatively graded polynomial ring if is regular.
{"title":"Elimination by substitution","authors":"Martin Kreuzer , Lorenzo Robbiano","doi":"10.1016/j.jsc.2025.102445","DOIUrl":"10.1016/j.jsc.2025.102445","url":null,"abstract":"<div><div>Let <em>K</em> be a field and <span><math><mi>P</mi><mo>=</mo><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>. The technique of elimination by substitution is based on discovering a coherently <span><math><mi>Z</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span>-separating tuple of polynomials <span><math><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> in an ideal <em>I</em>, i.e., on finding polynomials such that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>∖</mo><mi>Z</mi><mo>]</mo></math></span>. Here we elaborate on this technique in the case when <em>P</em> is non-negatively graded. The existence of a coherently <em>Z</em>-separating tuple is reduced to solving several <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-module membership problems. Best separable re-embeddings, i.e., isomorphisms <span><math><mi>P</mi><mo>/</mo><mi>I</mi><mo>⟶</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>∖</mo><mi>Z</mi><mo>]</mo><mo>/</mo><mo>(</mo><mi>I</mi><mo>∩</mo><mi>K</mi><mo>[</mo><mi>X</mi><mo>∖</mo><mi>Z</mi><mo>]</mo><mo>)</mo></math></span> with maximal #<em>Z</em>, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⟶</mo><mi>P</mi><mo>/</mo><mi>I</mi></math></span> as a fibration over an affine space, we show that its fibers allow optimal <em>Z</em>-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of <em>P</em> such that additional <em>Z</em>-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between <span><math><mi>P</mi><mo>/</mo><mi>I</mi></math></span> and a non-negatively graded polynomial ring if <span><math><mi>P</mi><mo>/</mo><mi>I</mi></math></span> is regular.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102445"},"PeriodicalIF":0.6,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768989","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}
Pub Date : 2025-02-27DOI: 10.1016/j.jsc.2025.102436
Guo-Shuai Mao , Xiran Zhang
In this paper, we mainly prove the following conjectures of Z.-W. Sun: Let be a prime. Then where stands for the Legendre symbol. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities.
{"title":"Proof of some conjectural congruences involving products of two binomial coefficients","authors":"Guo-Shuai Mao , Xiran Zhang","doi":"10.1016/j.jsc.2025.102436","DOIUrl":"10.1016/j.jsc.2025.102436","url":null,"abstract":"<div><div>In this paper, we mainly prove the following conjectures of Z.-W. Sun: Let <span><math><mi>p</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span> be a prime. Then<span><span><span><math><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</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><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mn>8</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mo>≡</mo><mo>−</mo><mrow><mo>(</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>)</mo></mrow><mfrac><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>4</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mspace></mspace><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>,</mo><mn>3</mn><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><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><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mn>8</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mo>≡</mo><mi>p</mi><mo>+</mo><mrow><mo>(</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>)</mo></mrow><mfrac><mrow><mn>2</mn><mi>p</mi></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>4</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mfrac><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mo>(</mo><mfrac><mrow><mo>⋅</mo></mrow><mrow><mi>p</mi></mrow></mfrac><mo>)</mo></math></span> stands for the Legendre symbol. 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":"130 ","pages":"Article 102436"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529701","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 : 2025-02-26DOI: 10.1016/j.jsc.2025.102435
Michael Burr, Michael Byrd
We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of algebraic curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main challenge in this algorithm is to correctly and efficiently identify and isolate all intersections between the curves. To overcome this challenge, we introduce a new and simple test that guarantees the global correctness of our output. A main step in our algorithm for approximating any number of curves is to correctly approximate a pair of curves. In addition to developing the details of this special case, we provide complexity analyses for both the number of steps and the bit-complexity of this algorithm using both worst-case bounds as well as those based on continuous amortization and condition numbers.
{"title":"Certified simultaneous isotopic approximation of algebraic curves via subdivision","authors":"Michael Burr, Michael Byrd","doi":"10.1016/j.jsc.2025.102435","DOIUrl":"10.1016/j.jsc.2025.102435","url":null,"abstract":"<div><div>We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of algebraic curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main challenge in this algorithm is to correctly and efficiently identify and isolate all intersections between the curves. To overcome this challenge, we introduce a new and simple test that guarantees the global correctness of our output. A main step in our algorithm for approximating any number of curves is to correctly approximate a pair of curves. In addition to developing the details of this special case, we provide complexity analyses for both the number of steps and the bit-complexity of this algorithm using both worst-case bounds as well as those based on continuous amortization and condition numbers.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"131 ","pages":"Article 102435"},"PeriodicalIF":0.6,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685764","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 : 2025-02-19DOI: 10.1016/j.jsc.2025.102434
Vaishak Belle
There is a fundamental trade-off between the expressiveness of the language and the tractability of the reasoning task in knowledge representation. On the one hand it is widely acknowledged that relations and more generally, the expressiveness of first-order logic is extremely useful for capturing concepts required for common-sense reasoning. But at the same time the entailment problem is only semi-decidable.
There have been a wide range of approaches to deal with this trade-off, from restricting the language to propositional logic to limit the expressiveness of the language in terms of the arity of the predicates (as in description logics) or the use of negation (as in Horn logic) to limit reasoning by weakening the entailment relation using non-standard semantics.
In this work, we address a gap in this literature. We show that there is an intuitive fragment of first-order disjunctive knowledge, for which reasoning is decidable and can be reduced to propositional satisfiability. Knowledge bases in this fragment correspond to universally quantified first-order clauses, but without arity restrictions and without restrictions on the appearance of negation. Queries, however, are expected to be ground formulas. We achieve this result by showing how the entailment over infinitely many infinite-sized structures can be reduced to a search over finitely many finite-size structures. The crux of the argument lies in showing that constants not mentioned in the knowledge base and/or query behave identically (in a suitable formal sense). We then go on to also show that there is also an extension to this result for function symbols.
{"title":"A propositional encoding for first-order clausal entailment over infinitely many constants","authors":"Vaishak Belle","doi":"10.1016/j.jsc.2025.102434","DOIUrl":"10.1016/j.jsc.2025.102434","url":null,"abstract":"<div><div>There is a fundamental trade-off between the expressiveness of the language and the tractability of the reasoning task in knowledge representation. On the one hand it is widely acknowledged that relations and more generally, the expressiveness of first-order logic is extremely useful for capturing concepts required for common-sense reasoning. But at the same time the entailment problem is only semi-decidable.</div><div>There have been a wide range of approaches to deal with this trade-off, from restricting the language to propositional logic to limit the expressiveness of the language in terms of the arity of the predicates (as in description logics) or the use of negation (as in Horn logic) to limit reasoning by weakening the entailment relation using non-standard semantics.</div><div>In this work, we address a gap in this literature. We show that there is an intuitive fragment of first-order disjunctive knowledge, for which reasoning is decidable and can be reduced to propositional satisfiability. Knowledge bases in this fragment correspond to universally quantified first-order clauses, but without arity restrictions and without restrictions on the appearance of negation. Queries, however, are expected to be ground formulas. We achieve this result by showing how the entailment over infinitely many infinite-sized structures can be reduced to a search over finitely many finite-size structures. The crux of the argument lies in showing that constants not mentioned in the knowledge base and/or query behave identically (in a suitable formal sense). We then go on to also show that there is also an extension to this result for function symbols.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102434"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474775","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}
Pub Date : 2025-02-19DOI: 10.1016/j.jsc.2025.102432
Hao Du , Clemens G. Raab
In symbolic integration, the Risch–Norman algorithm aims to find closed forms of elementary integrals over differential fields by an ansatz for the integral, which usually is based on heuristic degree bounds. Norman presented an approach that avoids degree bounds and only relies on the completion of reduction systems. We give a formalization of his approach and we develop a refined completion process, which terminates in more instances. In some situations when the completion process does not terminate, one can detect patterns allowing to still describe infinite reduction systems that are complete. We present such infinite systems for the fields generated by Airy functions and complete elliptic integrals, respectively. Moreover, we show how complete reduction systems can be used to find rigorous degree bounds. In particular, we give a general formula for weighted degree bounds and we apply it to find tight bounds in the above examples.
{"title":"Reduction systems and degree bounds for integration","authors":"Hao Du , Clemens G. Raab","doi":"10.1016/j.jsc.2025.102432","DOIUrl":"10.1016/j.jsc.2025.102432","url":null,"abstract":"<div><div>In symbolic integration, the Risch–Norman algorithm aims to find closed forms of elementary integrals over differential fields by an ansatz for the integral, which usually is based on heuristic degree bounds. Norman presented an approach that avoids degree bounds and only relies on the completion of reduction systems. We give a formalization of his approach and we develop a refined completion process, which terminates in more instances. In some situations when the completion process does not terminate, one can detect patterns allowing to still describe infinite reduction systems that are complete. We present such infinite systems for the fields generated by Airy functions and complete elliptic integrals, respectively. Moreover, we show how complete reduction systems can be used to find rigorous degree bounds. In particular, we give a general formula for weighted degree bounds and we apply it to find tight bounds in the above examples.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102432"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509115","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 : 2025-02-19DOI: 10.1016/j.jsc.2025.102431
Olivia X.M. Yao, Xuan Yu
Recently, Dockery, Jameson, Sellers and Wilson introduced new combinatorial objects called d-fold partition diamonds, which generalize both the classical partition function and the plane partition diamonds of Andrews, Paule and Riese. They also investigated a partition function which counts the number of Schmidt type d-fold partition diamonds of n. They presented the generating functions of and proved several congruences for . At the end of their paper, they posed a conjecture on congruences modulo 7 for and . In this paper, we prove the conjectural congruences for by using two methods: an elementary proof based on a result of Garvan and an algorithmic proof based on the Mathematica package RaduRK by Smoot. We also give an algorithmic proof of the conjectural congruences for by utilizing Smoot's Mathematica package RaduRK. In addition, we prove new infinite families of congruences modulo 7 for and prove that takes integer values with probability 1 for . Moreover, we show that there exist infinitely many integers such that with .
{"title":"Congruence properties for Schmidt type d-fold partition diamonds","authors":"Olivia X.M. Yao, Xuan Yu","doi":"10.1016/j.jsc.2025.102431","DOIUrl":"10.1016/j.jsc.2025.102431","url":null,"abstract":"<div><div>Recently, Dockery, Jameson, Sellers and Wilson introduced new combinatorial objects called <em>d</em>-fold partition diamonds, which generalize both the classical partition function and the plane partition diamonds of Andrews, Paule and Riese. They also investigated a partition function <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> which counts the number of Schmidt type <em>d</em>-fold partition diamonds of <em>n</em>. They presented the generating functions of <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and proved several congruences for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. At the end of their paper, they posed a conjecture on congruences modulo 7 for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. In this paper, we prove the conjectural congruences for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by using two methods: an elementary proof based on a result of Garvan and an algorithmic proof based on the Mathematica package RaduRK by Smoot. We also give an algorithmic proof of the conjectural congruences for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by utilizing Smoot's Mathematica package RaduRK. In addition, we prove new infinite families of congruences modulo 7 for <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and prove that <span><math><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>6</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mn>7</mn><mi>n</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>7</mn></mrow></mfrac></math></span> takes integer values with probability 1 for <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>. Moreover, we show that there exist infinitely many integers <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> such that <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>12</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>≡</mo><mspace></mspace><mi>i</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>13</mn><mo>)</mo></math></span> with <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>12</mn></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102431"},"PeriodicalIF":0.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509114","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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