polynomial identity testing

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 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 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 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 work addresses identity testing for arithmetic circuits containing exponential gates (x ↦ eˣ), i.e., deciding whether two such circuits compute the same real-valued function—specifically, functions of the form P(𝐱)/P′(𝐱), where P and P′ are exponential polynomials. Method: We propose the first efficient randomized black-box identity testing algorithm over finite fields for this class. We formally define the black-box model for exponential circuits and, assuming the Generalized Riemann Hypothesis (GRH), achieve perfect completeness and high-probability soundness; the false positive rate is exponentially suppressible. Contribution/Results: Our algorithm is implemented in Mirage—a compiler presented at OSDI ’25—and deployed in neural network optimization. Empirical evaluation demonstrates significant improvements in transformation correctness and compilation speed, with low overhead, strong robustness, and practical effectiveness.

This work investigates the problem of efficiently testing whether an unknown Boolean function is an s-term DNF formula under the relative error model, with particular focus on functions that depend on super-constantly many variables. The authors propose a tester with query complexity poly(s, 1/ε), which is independent of the total number of variables n. The key innovation lies in the novel decomposition of s-term DNFs into “local clusters,” enabling the development of structural analysis and testing techniques applicable even when the DNF representation is not explicitly given. This paper establishes, for the first time, that a natural class of Boolean functions—namely, s-term DNFs depending on super-constantly many variables—admits efficient testing in the relative error model with query complexity depending only on s and 1/ε.

NLSAT solvers suffer from low efficiency in constructing sign-invariant connected cells for nonlinear real arithmetic (NRA) satisfiability checking, as generating such cells incurs high computational cost due to algebraic complexity. Method: We propose a dynamic “trading quantity for speed” optimization strategy: adaptively introducing a small number of auxiliary linear polynomials during cell construction to reduce the algebraic complexity of individual cells. Our approach integrates symbolic-computation-driven sign-invariance analysis, Boolean-reasoning-guided polynomial expansion heuristics, and seamless integration into the NLSAT framework. Contribution/Results: The method is theoretically guaranteed to preserve completeness and correctness. Experiments show substantial reductions in both cell representation size and construction time; on standard NRA benchmarks, average solving speed improves by 32%, empirically validating the effectiveness of the complexity–quantity trade-off.

This work addresses the exponential blowup arising from structural mismatches in conversions between conjunctive normal form (CNF) and algebraic normal form (ANF) by introducing power-term polynomial algebra as a novel intermediate representation. The proposed framework seamlessly integrates CNF clauses with structured monomials, enabling symbolic manipulation without introducing auxiliary variables or expanding into conventional ANF. By defining power terms, local rewriting rules, and a systematic product reconstruction mechanism, it establishes a unified formalism that preserves the compactness of CNF while supporting the algebraic operability of ANF. The resulting canonical representation compactly encodes disjunctive clauses and facilitates efficient local transformations, offering a new paradigm for hybrid reasoning and structure-aware bidirectional conversion between CNF and ANF.

This work addresses the absence of deterministic polynomial-time identity testing (PIT) and reconstruction algorithms for depth-4 arithmetic circuits with unbounded top fan-in and high-degree powering gates. Focusing on the generalized Waring decomposition model (Σ∧ΣΠ)—sums of high powers of low-degree sparse polynomials—the paper presents the first deterministic PIT and reconstruction algorithms for such depth-4 power circuits where both the top fan-in and bottom degree may grow polynomially, without relying on non-degeneracy or average-case assumptions. The approach leverages the function-field ABC theorem, Wronskian differential operators and their kernel structure, and a robust variant of the Klivans–Spielman hitting set. For $d > r^2$, an explicit hitting set of size $O(r^4 s^4 n^2 d \delta^3)$ is constructed; when $d = \Omega(r^4 \delta)$, reconstruction is achievable in $\mathrm{poly}(n, s, d)$ time over fields of characteristic zero or sufficiently large characteristic.