polynomial identity testing

This work addresses the Polynomial Identity Testing (PIT) problem in non-associative polynomial algebras. Motivated by the longstanding lack of efficient algorithms for PIT in this setting, we establish the first Amitsur–Levitzki-type theorem for non-associative algebras and construct canonical algebraic models free of low-degree identities. Leveraging these structural results, we design a randomized polynomial-time black-box PIT algorithm and a deterministic polynomial-time white-box PIT algorithm. Furthermore, we construct a quasipolynomial-size hitting set for constant-depth non-associative arithmetic circuits. Our results break fundamental complexity barriers for PIT in the non-associative regime, resolving several long-standing open problems. They also introduce a novel computational framework for modeling non-associative structures within algebraic complexity theory, thereby advancing the theoretical foundations of non-associative computation.

This paper resolves the black-box polynomial identity testing (PIT) problem for noncommutative polynomials computed by arbitrary constant-depth $+$-regular circuits, breaking the long-standing depth-3 restriction of [AJMR, STOC’17/ToC’19]. We introduce novel techniques—including nondeterministic substitution automata, commutativization transforms, matrix algebra embeddings, and dynamic coefficient modulation—to design the first efficient randomized black-box PIT algorithm for constant-depth $+$-regular circuits, running in time $s^{O(d^2)}$, where $s$ is the circuit size and $d$ is the depth. Our key theoretical contribution is a tight matrix-size bound: any nonzero noncommutative polynomial vanishes identically over all $N imes N$ matrices only if $N < s^{O(d^2)}$; equivalently, every such polynomial evaluates to a nonzero matrix for some $N imes N$ matrix assignment with $N = s^{O(d^2)}$. This yields an optimal upper bound on the required matrix dimension for black-box PIT, enabling efficient evaluation-based testing.

This work addresses the challenge of efficiently encoding and verifying mathematical proofs in zero-knowledge settings. We propose a structured representation based on $2 imes2$ matrix-valued polynomials: formal statements are mapped to multivariate polynomial matrices with integer coefficients, and compact numeric fingerprints are generated via random evaluation over a finite field; matrix homomorphism enables algebraic execution of logical operations—including hypothetical reasoning and variable substitution. Our key contribution is the first encoding of formula syntactic structure into computable polynomial-matrix “fingerprints”, supporting incremental, zero-knowledge proof derivation and verification. Experiments demonstrate that only the fingerprints of initial axioms require explicit verification; all subsequent formulas are derived efficiently from ancestral fingerprints, drastically reducing both communication and computational overhead in verification.

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 derandomization of Polynomial Identity Testing (PIT). We introduce a novel hitting-set generator based on low-degree univariate rational function evaluation, leveraging coordinate-wise variable association for efficient distinguishability. First, we establish scaling equivalence between rational-function-based hitting sets and the Shpilka–Volkovich generator. Second, we systematically characterize their vanishing ideal, deriving tight bounds on minimal degree, sparsity, and multilinear partition complexity. Third, we design a deterministic membership test via alternating algebra. Our framework uniformly reconstructs classical PIT derandomization results and—crucially—extends them to read-once oblivious algebraic branching programs (ROABPs), achieving both efficient derandomization and new lower-bound proofs. The approach bridges rational-function interpolation, algebraic geometry, and structural complexity, yielding improved conceptual clarity and broader applicability in arithmetic circuit complexity.

This work addresses the limitation in Lean 4 that its native multivariate polynomials employ a non-computable representation, hindering efficient execution of symbolic computations such as Gröbner basis algorithms and thus impeding large-scale algebraic reasoning. To overcome this, the authors propose a certificate-based hybrid verification framework that integrates a computable polynomial representation, leverages external computer algebra systems (e.g., SageMath, SymPy) to compute Gröbner bases, and formally verifies their correctness within Lean 4. This approach enables, for the first time, practical automated handling of large-scale polynomial problems in Lean 4, supporting essential reasoning tasks including remainder verification, ideal membership testing, and ideal equality. The method substantially enhances both the efficiency and practicality of formalized algebraic reasoning in proof assistants.

This work addresses the poor scalability of traditional symbolic methods in automatically proving high-dimensional polynomial inequalities by introducing NSPI, a novel neurosymbolic framework that deeply integrates large language models (LLMs), symbolic computation, and formal verification in Lean. The approach leverages an LLM to generate sum-of-squares (SOS) decomposition conjectures, which are then refined into exact representations through symbolic optimization and formally verified within Lean, thereby establishing an end-to-end pipeline from heuristic discovery to rigorous proof. By synergistically combining the exploratory power of neural models with the mathematical rigor of symbolic and formal methods, NSPI achieves significantly improved efficiency and scalability, demonstrating strong performance on benchmarks involving up to ten variables.

This work addresses the central open problem of deterministic black-box polynomial identity testing (PIT) for read-once oblivious algebraic branching programs (ROABPs) with unknown variable order. It introduces an “algebraic rigidity” framework that constructs deterministic witnesses by exploiting the low-dimensional freedom of the matrix-word algebra associated with ROABPs, without relying on variable ordering or combinatorial isolation. The proposed method yields the first order-oblivious deterministic black-box PIT algorithm with running time \( n \cdot (wd)^{O(w^2)} \); under the modular stability conjecture, this complexity can be further improved to fully polynomial time. The paper also introduces novel tools—including algebraic rigidity, intrinsic invariants, and cyclic quotient ring hashing—offering fresh perspectives in algebraic complexity theory.

This work addresses the computational inefficiency caused by symmetry in integer programming, which leads to redundant search. The authors propose an algebraic approach that leverages arbitrary basis polynomials and problem-specific permutation groups to automatically generate random polynomial inequalities for symmetry breaking. This method constitutes the first general-purpose framework for generating symmetry-breaking constraints grounded in algebraic structure, applicable to any permutation group and seamlessly integrable into mainstream symbolic computation platforms. Experimental results on nearly half-capacity 0–1 bin packing instances demonstrate that incorporating only a small number of quadratic symmetry-breaking constraints significantly reduces solution time, with the most consistent performance gains observed when combining limited variables and permutations.

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.