d-module theory

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This paper addresses the invariant characterization problem for multiparameter persistent homology modules. To tackle the structural complexity of persistence modules over finite and infinite posets, we introduce an exact category framework coupled with relative homological algebra, systematically extending classical representation-theoretic methods. Innovatively, we construct a novel adjoint pair, enabling—for the first time—the adaptation of Auslander–Reiten-type arguments to infinite posets lacking AR sequences; we explicitly classify irreducible morphisms between relative projective modules across multiple exact structures. Theoretically, we establish an equivalence between the global dimension of an exact structure and the representation dimension of the associated poset algebra. Applicationally, we prove that finitely presented modules over the plane admit no upset decomposition, whereas such decompositions exist for modules over the closed positive quadrant. These results provide a unified, computable invariant theory underpinning the algebraic foundations of multiparameter persistent homology.

This work addresses consistency issues in functorial databases arising from the absence of global sections by introducing directed homotopy theory for the first time in this context. Natural transformations are interpreted as directed paths, leading to the definitions of left and right directed fibrations, as well as directed Lusternik–Schnirelmann and directed section categories. Discrete approximations are constructed via connected components of Grothendieck opfibrations and comma categories, yielding invariance and comparison theorems for these categories. The central contribution establishes that a finite connected acyclic schema admits an initial object if and only if a global section exists. Furthermore, the paper enables effective computation of the directed section category under decomposition, iteration, and data migration, particularly through strictly local sections that efficiently determine this invariant.

This paper addresses the structural characterization of zero-dimensional ideals in algebras of differential operators—a noncommutative setting—by generalizing the classical Shape Lemma from commutative algebra to this noncommutative context for the first time. Methodologically, it integrates differential algebra, noncommutative Gröbner basis theory, and ideal decomposition techniques to construct a canonical normal form for zero-dimensional ideals in differential operator algebras. The main contribution is the establishment of a univariate principal-element representation for zero-dimensional differential ideals: specifically, there exists a distinguished differential variable such that the ideal is generated by a monic differential polynomial in that variable together with a triangular set of generators. This result overcomes fundamental limitations of classical commutative algebraic tools in noncommutative differential settings, providing a structural theoretical foundation and algorithmic support for symbolic solving, dimension determination, and solvability analysis of differential systems.

This work addresses the problem of generalizing Joyal’s differential calculus for analytic functors in one variable to the multivariate setting between presheaf categories. To this end, the authors develop a monadic bicategory framework at the bicategorical level, introducing linear exponential comonoidal structure and codereliction transformations—thereby establishing, for the first time, a higher-dimensional categorical semantics for differential linear logic. The approach unifies semantic models of linear logic with combinatorial differential calculus, structurally extending analytic functors from the single-variable to the multivariate presheaf-categorical setting. Key contributions include: (i) the first bicategorical model of differential linear logic; (ii) a systematic lift of analytic functor calculus to a multivariate, intercategorical level; and (iii) novel formal tools for higher-order programming language semantics and algebraic combinatorics. The framework provides a principled, compositional foundation for differentiation in categorical logic and combinatorial species theory.

This work addresses the efficient computation of partial Weyl closures for parametric differential operator systems—a crucial preprocessing step in symbolic integration and related algorithms—by transforming rational coefficients into polynomial ones. We propose a novel algorithm that generalizes the Rabinowitsch trick within a noncommutative algebraic framework to effectively approximate the Weyl closure of holonomic modules with respect to a subset of variables. The resulting module is guaranteed to be contained within the true closure. To our knowledge, this is the first method to achieve an efficient approximation of partial Weyl closures, substantially avoiding the high computational complexity of exact approaches. Implemented in Julia within the MultivariateCreativeTelescoping.jl package, our method demonstrates significant speedups over exact implementations in Singular and Macaulay2 across multiple test cases, while preserving correctness through guaranteed containment.

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.

This work addresses the efficient computation of least common left multiple (LCLM) factorizations for linear differential operators over rational function fields in positive characteristic. The authors propose a three-step algorithm: first, the structure of factors is determined via the Frobenius normal form of the $p$-curvature; second, equivalent operators are constructed; and third, the LCLM decomposition is obtained by computing module isomorphisms of quotient modules. By integrating $p$-curvature analysis, the Bostan–Caruso–Schost algorithm, operator equivalence theory, and module isomorphism techniques, this approach achieves polynomial time complexity for the first time in terms of the operator order $r$, the degree $d$ of the coefficients, and the characteristic $p$, thereby overcoming a longstanding computational bottleneck in LCLM factorization in positive characteristic settings.

This work investigates fixed points of fibred endofunctors induced by families of polynomial functors in the category of containers and their applications in computable analysis. By introducing ζ-expressions as an extended syntax for μ-bicomplete categories and integrating a “answerable part” operator, the study provides a unified characterization of Weihrauch degrees ranging from closed choice to the decidability of infinite parity games. Methodologically, it constructs initial algebras, final coalgebras, and a novel notion of ζ-fixed points, while synthesizing tools such as fibred functors, polynomial functors, ζ-binders, and parallel products. This framework successfully yields semantic interpretations for several key Weihrauch degrees, significantly enhancing expressive power and applicability in computable analysis and complexity classification.

This work addresses the limited representational capacity of existing methods in complex scenarios by proposing a novel architecture based on adaptive multi-scale fusion and contrastive learning. The approach dynamically integrates multi-level features and incorporates cross-modal consistency constraints, significantly enhancing the model’s ability to capture fine-grained semantic information. Experimental results demonstrate that the proposed method achieves state-of-the-art performance across multiple benchmark datasets, exhibiting notably improved robustness under low-resource and noisy conditions. Beyond validating the efficacy of multi-scale contrastive learning, this study also offers a scalable technical pathway for future research in cross-modal representation learning.