algebraic number theory

This study investigates the algebraicity and arithmetic properties of hypergeometric functions over the rational numbers, finite fields, and p-adic fields. Leveraging the SageMath computer algebra system, the work integrates techniques from algebraic number theory, finite field theory, and p-adic analysis to systematically implement, for the first time in an open-source framework, algorithms capable of determining algebraicity, computing valuations, and solving for minimal polynomials in positive characteristic. This implementation fills a critical gap in existing computational toolchains by enabling uniform arithmetic analysis of hypergeometric functions across multiple number-theoretic domains, thereby substantially enhancing SageMath’s capacity for algebraic manipulation of such functions.

Computations of fundamental invariants of integer rings of number fields—such as integral bases and discriminants—lack formal verification, undermining their reliability in algebraic number theory. Method: We develop the first end-to-end formally verified computational framework for these invariants in Lean 4. Our approach comprises: (i) designing computable data structures for algebraic numbers and ideals; (ii) formally verifying key algebraic tools—including resultants, discriminants, and irreducibility tests for polynomials over ℚ and finite fields; and (iii) integrating Lean 4’s theorem prover with the SageMath computer algebra system via a certified interface. Contribution/Results: This work delivers the first fully formalized and machine-checked pipeline for computing integral bases and discriminants. We successfully verify these invariants for multiple number fields from the LMFDB, producing certificates checkable by Lean’s kernel. The framework establishes a trusted foundation for computational algebraic number theory and advances the synergistic integration of formal mathematics and computer algebra.

UniMath lacks computational support for algebraic structures—particularly W-types—and syntactic term algebras. Method: We develop a general algebraic library formalizing multi-sorted signatures and equational systems; implement, for the first time in UniMath, a computationally tractable subclass of W-types; construct, via explicit categorical methods, the univalent category of single-sorted ground term algebras; and rigorously prove its universality as the initial object in the category of algebras. We further establish an equivalence between ground term algebras and homotopical W-types. Contribution/Results: The library enables computable term algebras without recourse to general inductive definitions. We validate its effectiveness on foundational examples from universal algebra and propositional logic. This work substantially extends UniMath’s capacity to formalize and compute with structured mathematical objects, bridging abstract algebraic semantics and homotopy-theoretic type theory.

This study investigates the arithmetic properties of hypergeometric functions over the p-adic numbers, with a focus on their p-adic valuations and reduction behavior modulo primes. Building upon Christol’s theorem and integrating p-adic analysis with algebraic algorithms, the work achieves the first exact computation of p-adic valuations within arbitrary disks of convergence and establishes a systematic, effective criterion for determining the mod-p reducibility of hypergeometric functions. Furthermore, it introduces an algorithm to construct annihilating polynomials for the reductions modulo p. These contributions provide practical computational tools for the theory of arithmetic D-modules and significantly advance the algorithmic understanding of the arithmetic properties of hypergeometric functions.

This work addresses the satisfiability checking and quantifier elimination problems for nonlinear real arithmetic (NRA) formulas featuring both Boolean structure and quantifiers. We present the first extension of cylindrical algebraic coverings (CAC) to full first-order logic formula verification. Our approach introduces a novel CAC variant that integrates CAD-based covering construction, hierarchical quantifier handling, explicit Boolean structure incorporation, and adaptive splitting and pruning heuristics. Unlike conventional methods, our framework avoids constructing a complete cylindrical algebraic decomposition (CAD), thereby substantially reducing computational complexity. Experimental evaluation on diverse nonlinear quantified benchmarks demonstrates that our method outperforms state-of-the-art SMT solvers—including Z3 and CVC5—as well as specialized quantifier elimination tools such as QEPCAD and Redlog, in both solution accuracy and runtime efficiency. The gains are particularly pronounced on high-dimensional, sparse constraint instances.

This work addresses the problem of efficiently computing deterministic two-element representations of ideals in number fields. Focusing on ideals whose norm is coprime to the index of the defining polynomial’s ring of integers—a class that includes cryptographically relevant cases such as those defined by cyclotomic polynomials—the paper presents the first deterministic polynomial-time algorithm for this task. The approach leverages a generalized Dedekind criterion to decompose and construct ideals within number fields defined as ℚ[x]/(f). This method overcomes prior limitations that relied on randomization or failed to scale to cryptographic parameters, thereby achieving, for the first time, a combination of determinism, efficiency, and completeness across a broad and practically significant class of ideals used in cryptographic applications.

This work addresses the longstanding absence of formal verification for fundamental invariants in computational algebraic number theory—such as discriminants, signatures, unit groups modulo $p$-th powers, and class groups—which has undermined the reliability of databases and computer algebra systems. We present the first comprehensive formal verification framework in Lean 4 that encompasses these invariants, integrating real closed field theory, subresultant sequences, and certified ideal arithmetic. By leveraging SageMath to automatically generate verifiable proof certificates, our approach achieves the first formal verification of signatures, unit groups modulo $p$-th powers, and class groups for high-degree number fields. This significantly enhances both the efficiency and trustworthiness of verifying critical invariants like discriminants, and has already enabled the certification of hundreds of number field entries in the LMFDB.

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 article focuses on some rings of integers of number fields which are known to be norm-Euclidean domains, but for which no explicit algorithm computing the Euclidean division has yet been studied or implemented. The rings of integers we are interested in were proven to be Euclidean by H.W. Lenstra, Jr in 1978; they include the $n$-th cyclotomic rings for $n=15,20,24$. We present an algorithm performing Euclidean division in these rings based on Lenstra's proof and a closest vector computation by Conway and Sloane, and study its complexity. We give a complete implementation of the algorithm in SageMath. We also estimate the size of the remainders obtained when computing Euclidean divisions with this algorithm.

This work addresses the lack of systematic, automated solvers for families of zero-dimensional radical ideals involving algebraically independent parameters by introducing the EliminationTemplates package in Macaulay2. The package provides the first implementation within Macaulay2 of a general framework for constructing and specializing elimination templates, integrating elimination theory, Gröbner bases, and parameter specialization techniques. By extending solver methodologies originally developed in computer vision to broader algebraic contexts, it enables the creation of reusable, automated solvers. The effectiveness and practicality of this approach are demonstrated through successful applications to multiple computer vision problems as well as other algebraic scenarios.