compute normal form

This work proposes an efficient method for computing Rational Univariate Representations (RUR) of zero-dimensional polynomial systems by leveraging dense linear algebra and Gaussian elimination. Building upon classical FGLM-type algorithms, the approach replaces conventional steps with Gaussian elimination, thereby significantly enhancing computational efficiency for large-scale systems while rigorously preserving theoretical correctness. Experimental results demonstrate that the proposed method correctly parameterizes zero-dimensional ideals with thousands of solutions in just a few seconds. The implementation is publicly available as the open-source Julia package RationalUnivariateRepresentation.jl.

This paper addresses the open problem of synthesizing guarded loops with polynomial invariants for formal verification. Prior approaches are limited to affine, unguarded loops; ours is the first algebraic-geometry-driven method supporting nondeterministic branching loops with inequality guards, polynomial update maps, and arbitrary polynomial invariants. Our core contributions are threefold: (1) introducing a novel class of synthesizable invariants; (2) reducing loop synthesis to solving systems of multivariate polynomial equations over the rationals; and (3) integrating algebraic-geometric techniques—such as Gröbner bases and elimination theory—with SMT solvers to efficiently compute solution spaces. We implement a prototype system and evaluate it on multiple benchmarks. Experimental results demonstrate its ability to synthesize finite-loop programs satisfying complex polynomial invariants, confirming correctness, effectiveness, and scalability.

This work addresses the challenge of synthesizing loop structures from polynomial invariants in program verification. We propose the first general loop synthesis method supporting loops with inequality guards, polynomial update maps, and arbitrary polynomial invariants—extending beyond the affine restrictions of prior approaches. Our method pioneers the application of algebraic geometry to loop synthesis: it reduces the synthesis problem to a decidable geometric formulation by constructing a finite system of polynomial equations whose complex solution set precisely characterizes all valid loops satisfying the given invariant. The algorithm integrates symbolic polynomial system solving with SMT reasoning (e.g., Z3) and is implemented and validated in a prototype tool. Our key contribution is the principled, provably sound, and computationally effective synthesis of non-affine loops—breaking the long-standing affine barrier and enabling structured, verifiable, and algorithmic generation of general polynomial loops.

This work addresses the lack of efficient parametric solution methods for zero-dimensional parametric polynomial systems that exhibit favorable specialization properties. It presents the first systematic study of the specialization behavior of Rational Univariate Representations (RURs) in parametric settings, establishing explicit upper bounds on the degree and height of their constituent elements. By leveraging techniques from algebraic geometry and symbolic computation, the authors develop a general RUR-based parametrization framework and introduce two efficient algorithms. The proposed approach guarantees stable specialization, provides rigorous algebraic complexity bounds, and yields a fully computable and implementable parametric solution method.

For zero-dimensional polynomial systems, this paper proposes a separation-based linear-type unconditional certification method and an algorithm for constructing Rational Univariate Representations (RURs) without requiring additional structural assumptions—such as the shape lemma. The core method derives the RUR directly from a lexicographic Gröbner basis via binary elimination ideals, augmented by Gaussian elimination and ideas inspired by the FGLM algorithm. This approach eliminates reliance on geometric properties of the ideal (e.g., radicality or generic position), ensuring both formal correctness and certifiability. Experimental results demonstrate high efficiency: a Maple implementation requires only ~300 lines of code, while a naive Julia implementation achieves state-of-the-art performance, significantly outperforming existing uncertified parametric solvers.

This work proposes an efficient algorithm for computing the leading principal submatrices of the Hermite normal form (HNF) of polynomial matrices endowed with displacement structure, addressing the high computational cost associated with full HNF computation. By integrating displacement structure with an evaluation–interpolation framework, the method recovers selected rows from the matrix inverse and derives the target submatrix via relation bases. This study presents the first fusion of displacement structure techniques with the evaluation–interpolation paradigm, achieving a significant reduction in computational complexity compared to conventional full HNF algorithms. The approach is particularly well-suited for applications involving Gröbner bases and bivariate polynomials, where structured polynomial matrices commonly arise.

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 proposes a systematic framework to enhance the computational efficiency and numerical stability of evaluating high-degree matrix polynomials. Specifically, for polynomial degrees eight and higher, the method generates and validates stable coefficient sets that reduce the number of required matrix multiplications by one compared to the classical Paterson–Stockmeyer scheme. To address instability issues in the original formulation, the authors introduce structural variants and design a reliability metric to assess the expected numerical accuracy of candidate coefficient sets. Nonlinear polynomial systems are solved using variable-precision arithmetic (VPA), and an in-house tool, MatrixPolEval1, enables efficient screening and validation. Applied to matrix exponentials and geometric series, the approach achieves a saving of one matrix multiplication while maintaining comparable numerical accuracy.

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.