hilbert series computation

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.

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.

Computing discrete cubic homology of graphs over finite fields suffers from low efficiency due to high-dimensional chain complexes, slow generation of singular cubes, and substantial redundant computation. To address this, we propose three innovations: (1) dimension reduction via quotient spaces constructed from cube automorphisms; (2) an efficient singular cube generation strategy that avoids redundant enumeration; and (3) an axiomatic graph preprocessing framework that eliminates topologically irrelevant substructures prior to homology computation. Our approach integrates algebraic topology, linear algebraic reduction over characteristic-zero fields, and graph-theoretic algorithms. Experiments demonstrate significant reductions in both time and space complexity: homology group computation accelerates by one to two orders of magnitude. The method enables scalable topological data analysis for large-scale graphs, providing a novel, efficient tool for discrete cubic homology computation.

Persistent homology suffers from computational inefficiency in computing interval bases of persistent modules, primarily due to reliance on matrix representations and sequential Smith normal form algorithms. To address this, we propose a presentation-free parallel decomposition framework that constructs interval generators via kernel-flag chains derived from structural maps—bypassing explicit matrix representation entirely and enabling scalable parallelization in distributed-memory environments. Theoretically, our approach unifies insights from linear algebra, representation theory, and Hodge decomposition; practically, it significantly accelerates interval basis extraction in dynamic topological analysis tasks such as harmonic tracking, outperforming state-of-the-art Smith-based methods in empirical benchmarks. Our key contribution is the first kernel-chain-driven, presentation-free parallel paradigm for persistent module decomposition, overcoming fundamental scalability bottlenecks inherent in classical algebraic topology computations.

This work proposes a novel and efficient method for computing isolated regular solutions of multivariate polynomial systems with composable structure. By introducing variable substitutions, the original system is reduced to a lower-dimensional system in intermediate variables. The approach integrates a probabilistic symbolic homotopy algorithm, algebraic independence analysis, and the Chevalley–Shephard–Todd theorem to handle cases invariant under finite reflection groups. It is the first systematic exploitation of composability in polynomial systems to substantially reduce symbolic solving complexity. The algorithm achieves arithmetic complexity polynomial in both the input size and the number of solutions. Experimental results on benchmark systems—including those invariant under symmetric groups, hyperoctahedral groups, and exceptional reflection groups—demonstrate its superior computational efficiency.

This work addresses the reliable computation of Galois and monodromy groups for parametrized polynomial systems. To this end, it introduces a novel framework that integrates certified homotopy path tracking with homotopy graphs, enabling—for the first time—the rigorous numerical verification of monodromy group actions. By combining certified numerical algorithms with techniques from numerical algebraic geometry, the proposed method guarantees the mathematical correctness of its computational results. The approach has been successfully validated on a range of examples drawn from both pure and applied mathematics, demonstrating its effectiveness, reliability, and practical utility in analyzing the group-theoretic structures of complex polynomial systems.

This study investigates homological invariants of zero-divisor graphs over finite chain rings, with a focus on the Betti numbers and algebraic properties of their edge ideals. By constructing a layered graph \( C(q,L) \) that encodes the zero-divisor structure, the authors prove that this graph is cochordal and thereby determine its type sequence, leading to a corrected and refined formula for the Betti numbers of the associated edge ideal. Employing techniques from combinatorial commutative algebra, cochordal graph theory, and homological algebra, they compute the projective dimension and Castelnuovo–Mumford regularity of Gaussian quotient rings \( \mathbb{Z}_{2^m}[i] \) and truncated polynomial rings \( \mathbb{Z}_p[x]/(x^c) \), establishing that these rings admit 2-linear resolutions. Moreover, they show that such rings are Cohen–Macaulay only in degenerate or complete graph cases.

This work addresses the problem of efficiently computing the Gukov–Manolescu invariant from braid representations by proposing a three-stage approach: searching for suitable braids and reversal data, encoding the state space as a polyhedron under linear constraints, and constructing the invariant via products of R-matrices. The method enables, for the first time, scalable computation of this invariant and yields the first publicly available database. Integrating polyhedral encoding, systems of linear constraints, R-matrix operations, and optimized search algorithms, the framework supports computations for prime knots up to 12 crossings and prime links with up to 10 crossings and at most three components. Accompanying software includes a Python library, a command-line tool, and a Mathematica package.

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.