symmetry group characterization

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 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 paper addresses the problem of determining whether a given algebraic variety (V) arises as the Zariski closure of an orbit of a point under the action of an (s)-generated commutative matrix group. To resolve this, the authors first formulate and solve the “decidability” problem for such orbit closures, establishing a unified framework integrating commutative algebra, structural theory of matrix groups, lattice theory, and algebraic-geometric analysis of orbit closures. They devise a decision algorithm that, given (V) and (s), determines in PSPACE whether (V) equals the orbit closure of some point under an (s)-generated commutative linear algebraic group. Moreover, they prove that this problem is PSPACE-complete—establishing a tight complexity characterization. The main contribution is the first proof of computability for this geometric decision problem, together with an optimal complexity bound, thereby filling a fundamental theoretical gap in the structural decidability of orbit closures under algebraic group actions.

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 work addresses the problem of determining the continuous symmetries—specifically, the symmetry Lie algebra—of a parametrized algebraic variety directly from its parametric representation, without explicitly computing its vanishing ideal. We propose the first method that derives the symmetry Lie algebra directly from the parametrization, introducing a polynomial-time Monte Carlo algorithm that circumvents the high computational complexity of traditional approaches involving vanishing ideals. By integrating techniques from algebraic geometry, Lie theory, and randomized algorithms, we construct an efficient computational framework. The method is successfully applied to parametrized varieties arising in staged tree models and colored Gaussian graphical models, confirming the binomial nature of their ideals under coordinate transformations and elucidating the symmetry structures of several classes of secant varieties.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

In three-dimensional graphic statics, editing complex polyhedral graphs often compromises their symmetry, thereby undermining engineering applicability. This work introduces crystallographic point group theory into the field for the first time and establishes length consistency among equivalent edge sets as a necessary and sufficient condition for preserving symmetry. By integrating symmetry detection algorithms from spglib and pymatgen, the authors develop an efficient fingerprinting method to automatically classify equivalent edges and enforce corresponding constraints. Implemented in the PolyFrame 2 plugin, this approach significantly reduces the dimensionality of the solution space while effectively maintaining the symmetry of polyhedral graphs, thereby enhancing both design feasibility and computational efficiency.