field extension construction

This work addresses the problem of simplifying generating sets for subfields of multivariate rational function fields, aiming to construct smaller and more efficient generating sets from given complex ones. To this end, the authors propose a novel algorithm that integrates sparse interpolation with partial Gröbner basis computation, enabling efficient search for polynomials of fixed degree within the subfield while avoiding the high computational cost of full basis construction. The method significantly outperforms existing approaches in both efficiency and the quality of the resulting generating sets. Its effectiveness and practical utility are further demonstrated through applications in structural parameter identifiability, where the simplified generating sets yield meaningful and computationally advantageous representations.

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.

This work addresses the long-standing challenge of explicitly constructing near-optimal lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over small finite fields whose size depends only on the rank or codimension. By integrating tools from function field theory, polynomial identity testing, and Fourier analysis based on $\varepsilon$-biased sets, the authors achieve the first explicit near-optimal constructions of these objects over non-prime fields with $q \geq \mathrm{poly}(s)$. Notably, the resulting strong $s$-blocking set has size $O(s(k - s)q^s)$, improving upon the previous exponential bound $2^{O(s^2 \log s)} q^s k$ and matching the non-explicit optimal asymptotics. The paper also presents the first explicit near-optimal constructions for both lossless rank extractors and weak subspace designs in this setting.

This paper addresses the efficient computation of the characteristic polynomial of a $q$-linearized polynomial $L(x)$ of $q$-degree $r$ over the extension field $mathbb{F}_{q^n}$, where $L(x)$ is defined over the base field $mathbb{F}_q$. Traditional matrix-based approaches incur prohibitive complexity—either $O(n^3)$ or $O(n^omega)$ field operations—rendering them impractical for large $n$. To overcome this bottleneck, we propose the first quasi-square-root algorithm that exploits the intrinsic $q$-linearized structure of $L(x)$. Our method integrates divide-and-conquer recursion, fast polynomial arithmetic (leveraging FFT-based multiplication in $mathbb{F}_q[x]$), and key algebraic properties of $q$-polynomials. The resulting algorithm achieves a complexity of $O(n (log n)^4)$ $mathbb{F}_q$-operations. Crucially, this cost depends only on $q$ and $r$, not on $n$, thereby avoiding degradation with increasing extension degree. The approach yields both a theoretical breakthrough and practical acceleration, especially for low $q$-degree cases.

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.

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 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 lack of efficient algebraic modeling techniques for supersingular isogeny problems whose degrees are powers of 2 or 3. By leveraging Renes’ isogeny formulas in Montgomery form (for degree 2) or triangular form (for degree 3), the authors present the first reduction of these problems to zero-dimensional multivariate polynomial systems. They prove that the resulting systems are non-generic in their coordinate structure, thereby transcending the conventional modular polynomial framework. Through a careful analysis of the dimension of the highest-degree homogeneous components and the application of Gröbner basis techniques, the proposed approach demonstrates significantly superior performance over existing modular-polynomial-based algebraic solvers in experimental evaluations.