polynomial method

This work investigates the theoretical foundations and applicability limits of low-degree polynomial methods for characterizing average-case statistical-computational tradeoffs. Focusing on canonical statistical tasks—including detection and recovery—it proposes polynomial degree minimization as a complexity measure and develops a unified lower-bound toolkit integrating low-degree approximation, moment matching, high-dimensional probability, sum-of-squares (SoS) relaxations, and statistical query models. The paper provides the first systematic exposition of the framework’s validity conditions and conceptual underpinnings, elucidates its deep connections to SoS hierarchies and statistical query complexity, and establishes verifiable criteria for hardness certification. It precisely delineates both the explanatory power and intrinsic limitations of low-degree methods, while distilling key open questions. By doing so, it advances statistical computational complexity theory through a structured, principled analytical paradigm grounded in polynomial approximability.

This work investigates the closure of algebraic complexity classes—VP, VBP, and VNP—under polynomial factorization. Methodologically, it unifies the analysis of Hensel lifting and Newton iteration, precisely delineating their technical equivalence and applicability boundaries, while integrating tools from algebraic circuit complexity and formal power series expansions. The study systematically characterizes structural conditions and fundamental limitations for efficient factorization across these models. Key contributions include the first rigorous demonstration that VP is closed under factorization whereas VNP is not, thereby clarifying VBP’s intermediate status. Several pivotal open problems are posed, notably: “If $f in ext{VNP}$ and a factor $g$ of $f$ lies in VP, must $g$ have polynomially bounded VNP-circuit size?” These results establish a new theoretical framework and benchmark for the intersection of algebraic complexity theory and symbolic computation.

This paper addresses the connectivity determination problem for unbounded smooth real algebraic sets. We present the first Monte Carlo roadmap algorithm that does not require boundedness assumptions. Given an algebraic set defined by $n$-variate polynomials of degree at most $D$, the algorithm constructs a connectivity roadmap for any finite query point set, guaranteeing that the intersection of each connected component with the roadmap is itself connected. Technically, the method integrates critical point theory, stratified sampling, algebraic projection, and randomization. Its output size and running time are both bounded by $(nD)^{n log d}$, where $d$ is the maximum degree among input polynomials—constituting a fundamental improvement over the prior best bound of $(nD)^{n log^2 n}$. Moreover, the algorithm strictly ensures connectivity coverage for the given query points. To our knowledge, this is the first deterministic, verifiable roadmap construction achieving near-optimal algebraic complexity for unbounded real algebraic sets.

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.

This work investigates the computational complexity of algebraic attacks on the Learning With Errors (LWE) problem, focusing on Gröbner basis solvability of the Arora–Ge polynomial system. Methodologically, we first establish—rigorously and for the first time—that this system satisfies the generic coordinate condition introduced by Caminata–Gorla, thereby embedding it within a general algebraic framework. Second, we extend the Semaev–Tenti algorithm to arbitrary finite-regularity polynomial systems and, leveraging Castelnuovo–Mumford regularity and the Macaulay bound, derive the first precise subexponential upper bound on Gröbner basis computation complexity under the degree-reverse-lexicographic (DRL) monomial order—applicable to both generic and binary-secret/binary-error LWE instances. Finally, we propose a novel framework for incorporating side information (“hints”) into algebraic LWE modeling, enabling quantitative complexity assessment of attacks exploiting prior knowledge. These contributions provide both theoretical foundations and practical tools for algebraic security analysis of LWE.

Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation.

This work addresses the problem of computing sample points in each connected component of a semi-algebraic set defined by inequalities involving real-coefficient polynomials. Under generic smoothness assumptions on the input polynomials, the authors propose a probabilistic algorithm that characterizes connected components via critical points and reduces the problem to solving zero-dimensional polynomial systems. For the first time, the algorithm leverages the actual degree structure of both the input polynomials and their partial derivatives to deliver a refined bit complexity analysis based on Bézout bounds: parameterization of sample points achieves cubic complexity, while rational approximation incurs quartic complexity. Experimental results demonstrate that the method efficiently handles previously intractable instances, including random dense systems and those arising from practical applications.

This work addresses the maximization of multilinear polynomials over \( n \) binary variables, a problem that encompasses numerous NP-hard special cases, including unconstrained quadratic binary optimization. The authors introduce a general variable elimination framework that, for the first time, explicitly constructs an equivalent multilinear polynomial after elimination and enables efficient solution through recursive application. Built upon elementary algebraic operations, this approach unifies and extends all previously known tractable cases—such as those with bounded treewidth or various acyclic structures—yielding a broader and more computationally efficient algorithmic framework.

This work addresses a central barrier to resolving Valiant’s conjecture (VP ≠ VNP) by explicitly constructing elusive functions with prescribed parameters. By restricting coordinate mappings to monomials and leveraging hitting sets formed from roots of unity together with Chebotarev’s theorem, the construction of elusive functions is reduced— for the first time—to a problem of exponential sumset expansion in additive combinatorics. This approach not only resolves an open question posed by Garg et al. regarding the existence of elusive curves of exponential degree but also yields an explicit construction of such curves. Furthermore, for arithmetic circuits of depth below \(o(\log n / \log \log n)\), the method improves the best-known superlinear lower bound on input size to a quadratic one.