compute hilbert series

This work addresses the computational complexity of classical invariant theory in projective and enumerative geometry. To this end, we design and implement Brackets, the first open-source Macaulay2 package providing systematic support for bracket rings and Grassmann–Cayley algebras. The package introduces a declarative symbolic syntax tailored to SLₙ-invariants, integrates an efficient straightening algorithm, and—uniquely within Macaulay2—unifies core operations including bracket algebra, Plücker relation handling, and Schubert calculus. Applications include automated derivation and verification of fundamental projective-geometric statements such as cross-ratios, collinearity conditions, and intersection criteria. By enabling rigorous, symbolic reasoning over classical geometric invariants, Brackets significantly enhances both the discovery and computational efficiency of geometric theorems. This work fills a critical gap in computer algebra systems for automated invariant-theoretic reasoning in classical geometry.

This work proposes a novel representation learning framework based on adaptive multi-scale fusion and contrastive learning to address the limited representational capacity of existing methods in complex scenes. By dynamically integrating multi-granularity features and incorporating a structure-aware contrastive loss, the proposed approach effectively enhances the model’s ability to capture fine-grained semantics and contextual relationships. Extensive experiments demonstrate that the framework consistently outperforms state-of-the-art methods across multiple benchmark datasets, achieving substantial improvements in both accuracy and robustness. These results underscore its potential as a new technical pathway for tackling challenging visual understanding tasks.

This work addresses the computation of a homology basis for complex elliptic surfaces over ℙ¹, enabling period integrals and reconstruction of key algebraic invariants. We introduce the first systematic semi-numerical algorithm, implemented as an end-to-end framework in SageMath, integrating high-precision numerical integration, symbolic computation, and algebraic geometry theory. Our contribution is threefold: (i) we pioneer the application of semi-numerical techniques to construct homology bases for elliptic surfaces—overcoming limitations of purely symbolic or purely numerical approaches; (ii) we stably recover the Néron–Severi lattice, transcendental lattice, Mordell–Weil group, and its associated lattice structure; and (iii) the method significantly enhances the feasibility of period computations, providing a new computational paradigm for the classification of moduli spaces of elliptic surfaces and effective arithmetic-geometric calculations.

Existing algorithms for computing Betti tables and minimal presentations in zero-dimensional persistent homology suffer from prohibitively high computational complexity—up to $O(n^3)$—rendering them impractical for large-scale data. Method: We integrate multigraded commutative algebra, persistent homology theory, and graph/matrix optimization techniques, leveraging the intrinsic tree-like structure of zero-dimensional homology. Contribution/Results: We present two breakthroughs: (i) the first $O(n log n)$ log-linear-time algorithm for zero-dimensional Betti tables; and (ii) an $O(n^2)$ quadratic-time algorithm for minimal presentations under arbitrary poset gradings. Our Betti table computation achieves over three orders-of-magnitude speedup versus state-of-the-art methods, while minimal presentation computation is uniformly reduced to quadratic complexity. These advances significantly enhance the scalability and practicality of topological data analysis (TDA) in clustering, graph classification, and other large-scale applications.

This paper addresses the computation of β-graded vanishing ideals of subsets of toric varieties over finite fields—particularly weighted projective spaces—for constructing and analyzing toric codes. A key bottleneck is the lack of systematic methods for constructing vanishing ideals of rational point sets. We establish, for the first time, an intrinsic connection between such vanishing ideals and defining ideals of numerical semigroup rings. Leveraging this link, we propose a constructive generation algorithm based on subsemigroup structure, integrating tools from algebraic geometry, combinatorial commutative algebra, and Gröbner basis theory. Our method explicitly yields minimal β-graded generating sets for vanishing ideals in typical cases, significantly enhancing the algebraic modeling efficiency and parameter computability of toric codes. This advances the algebraic framework for error-correcting code design and provides a novel computational tool for toric coding theory.

This project aims to efficiently compute free resolutions of finitely generated modules over exterior algebras and the cohomology of coherent sheaves on projective spaces. Methodologically, it refines Schreyer's algorithm by introducing relative Gröbner bases and tree traversal techniques to resolve free decompositions over exterior algebras, while leveraging the Bernstein–Gel'fand–Gel'fand (BGG) correspondence to derive cohomology. Furthermore, parallelization strategies originally developed for polynomial rings are adapted to exterior algebras, establishing a large-scale parallel computing architecture. The project successfully achieves efficient computation of these algebraic objects, significantly enhancing solving performance. By validating the parallelization potential of exterior algebra computations, this work provides a novel acceleration paradigm for computational algebraic geometry.

This study investigates the graded Betti numbers and associated algebraic invariants of generalized split-join graph families. By decomposing the independence complex into an iterated join of disjoint unions of simplices and discrete complexes, and leveraging Hochster’s formula, the computation of Betti numbers is reduced to the explicit extraction of coefficients from generating functions. The work provides the first complete description of the graded Betti tables for generalized split graphs and clique-star graphs for arbitrary clique sizes, establishes a sharp criterion for 2-linear resolutions, and identifies the threshold at which the Castelnuovo–Mumford regularity stabilizes. Closed-form expressions are further derived for the linear strand, higher Betti numbers, Hilbert series, projective dimension, and extremal Betti numbers, with results extended to broader graph classes including pineapple graphs and multipartite powers.

This work presents the first complete formalization in Lean4 of the multigraded Proj construction introduced by Brenner–Schröer and its associated algebraic ring extensions, a theory long lacking rigorous machine-checked verification. Building upon the Mathlib library and grounded in dependent type theory and formal mathematics, the authors provide a fully verified construction of multigraded Proj schemes and their related ring-theoretic structures. This formalization not only fills a critical gap in the formalization of algebraic geometry in the multigraded setting but also substantially enhances the expressiveness and verifiability of the subject. The results lay a robust foundation for future formal developments in higher-dimensional algebraic geometry.

This work constructs an integer-valued Chern class for acyclic matroids and their Feichtner–Yuzvinsky wonderful compactifications, which in the realizable case corresponds to the Chern class of the tangent bundle of the associated wonderful compactification space. Leveraging the AI-driven mathematical reasoning agent Danus, this construction was achieved without prior human guidance, marking the first instance where an AI independently discovered and reproduced a key result later confirmed by human researchers. The approach synthesizes techniques from algebraic geometry, matroid theory, K-theory, and the Hirzebruch–Riemann–Roch theorem to carry out formal derivations. The resulting Chern class precisely recovers the Hilbert series of the Chow ring and satisfies the Chern-alpha lower bound, thereby demonstrating the capacity of artificial intelligence to achieve breakthroughs in cutting-edge problems of pure mathematics.

This work extends classical indicial polynomials and Bernstein–Sato polynomials to arbitrary subschemes, introducing for the first time a notion of indicial polynomial along an arbitrary subvariety within the framework of $D$-modules. This definition unifies the indicial polynomial of a single differential equation, the Bernstein–Sato polynomial of an algebraic variety due to Budur–Mustață–Saito, and the classical $b$-function along a smooth submanifold. Notably, it remains well-defined even when the $b$-function does not exist, and its set of roots recovers the $b$-function whenever the latter is defined. The approach combines $D$-module theory, algebraic geometry, and symbolic computation, leveraging inverse image functors and embedding techniques to handle parametric settings more efficiently. The result establishes a universal connection between indicial polynomials and $b$-functions, providing a more general and computationally tractable theoretical framework.