real algebraic geometry

This work addresses the intrinsic computational complexity of solving systems of algebraic equations. We introduce “Galois width” — a novel complexity measure quantifying solution difficulty within finite computational models restricted to field operations and polynomial root adjunctions. Methodologically, we integrate Galois theory, algebraic geometry, and monodromy group analysis to derive computable lower bounds on complexity and expose fundamental limitations of classical heuristics—such as monodromy continuation. Our principal contribution is the first Galois-structural complexity criterion for multisolvable algebraic problems arising in geometric modeling, optimization, statistical inference, and computer vision. This criterion enables principled algorithm selection: identifying low-width instances amenable to accelerated solving, or rigorously proving the inevitable failure of standard numerical or symbolic methods.

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 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 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 challenge of satisfiability in the existential theory of the reals (∃ℝ), particularly the realizability of configurations in discrete geometry, by introducing a novel paradigm termed Satisfiability Modulo Realizability. The approach encodes geometric problems as SAT instances over abstract order types and integrates diversity-driven sampling, partial realizability feedback, and a new flip-based heuristic to efficiently guide the solver away from non-realizable regions during search. Leveraging this framework, the study resolves a long-standing open problem in discrete geometry by proving that the largest point set containing no empty convex hexagon or convex heptagon has size 23, thereby substantially advancing the computational frontier in this domain.

This study investigates the decidability of the first-order theory of Presburger arithmetic extended with unary nonlinear predicates, such as fixed powers and polynomials of degree at most three. By integrating tools from algebraic geometry—specifically the theory of low-genus curves—with number-theoretic methods for solving Diophantine equations, the authors establish, for the first time, that this restricted nonlinear extension remains decidable. The work delineates a sharp boundary of decidability: even slight relaxations of the constraints lead to undecidability. Moreover, it reveals deep connections between more general extensions and classical open problems in Diophantine analysis, thereby underscoring both the tightness and theoretical significance of the obtained decidability result.

This work proposes a novel algorithm rooted in real algebraic geometry and symbolic computation to address the problem of semialgebraic description of connected components of real algebraic curves. The method efficiently characterizes each connected component by constructing Boolean formulas composed of polynomial equalities and inequalities with real coefficients. Compared to the current state-of-the-art approach based on isotopy graphs, the proposed algorithm achieves a significant reduction in computational complexity, thereby offering a notable breakthrough in efficiency. This advancement enables rapid analytical processing of real algebraic structures, effectively supporting practical applications such as optical system design and robotic path planning that demand timely and precise geometric reasoning.

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.