Open research questions in Polynomial and algebraic computation
81 gap statements mined from Polynomial and algebraic computation papers in our 4.5M-paper local library, which holds 494 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
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What the literature leaves open
The gap is the lack of effective constructions of geometric realizations of birational maps between Mori Dream Spaces. The gap is the lack of understanding of the properties of geometric realizations.
Existing solutions have limitations, such as targeting a single coordinate system or lacking extensibility. There is a need for a generic and extensible solution for geometric computations.
The lack of understanding of the Segre determinant and its applications. The need for a study of the properties of the Segre determinant. The need for an exploration of the applications of the Segre determinant to algebraic vision and Chow quotients of Grassmannians.
Future research can focus on improving the performance of the algorithms. Future research can explore the applications of the results in different fields.
The lack of an arithmetic for multidimensional continued fractions is a significant research gap. The prior work on continued fractions does not address the multidimensional case.
The arithmetic properties of multiple polylogarithms, such as irrationality and linear independence, have been extensively studied yet remain largely shrouded in mystery. There is a lack of criteria for the linear independence of multiple polylogarithm values over algebraic number fields.
Linear independence of periods related to polylogarithms · 2026The algorithm has a probabilistic nature, which may lead to failures. The algorithm is limited to the case where the hypersurface defined by the vanishing of the polynomial is smooth.
Computing points in connected components defined by a real inequation: algorithms, complexity and implementations, Part I · 2026The algorithm is limited to the special case of univariate polynomials in an Archimedean saturated quadratic module. The paper does not provide a general solution for all types of polynomials.
Computing Certificates in Archimedean Univariate Saturated Quadratic Modules · 2026Future research can focus on generalizing the algorithm to all types of polynomials. Future research can focus on improving the efficiency of the algorithm.
Computing Certificates in Archimedean Univariate Saturated Quadratic Modules · 2026The paper suggests a syzygy-theoretic approach to non-Desarguesian and non-Pappian geometries in higher ranks. Future research could explore the applications of this approach to other areas of mathematics and computer science.
Universal Incidence Syzygies: Bridging Desarguesian Embeddability and the Pappian Commutativity Constraint in Higher-Rank Projective Varieties · 2026 · DOIAs a direction for future research, adapting Newton’s method to operate in a manner similar to Lagrange interpolation may lead to a more efficient recovery algorithm (note that a multivariate extension of Newton’s method can be found in).
The lack of an unconditional geometric separation of VP and VNP. The reliance on global representation-theoretic weights or point-wise algebraic ideals in prior work.
A Five Part Validator-Grade Topological Closure and Localized Hessian Asymmetry: An Unconditional Geometric Separation of VP and VNP · 2026 · DOIComputing p-th roots of circuits: One of the simplest-to-state open problems in the area of factorization of algebraic circuits is the following — over a characteristic p field, if a polynomial f p is an n-variate, degree d polynomial computed by a poly(n, d)-sized circuit, is f also computable by a poly(n, d)-sized circuit? The answer to this question is unknown even for the setting of general algebraic circuits.
Future research can focus on developing new card games with unique mathematical properties. Future research can focus on creating educational materials for teaching linear algebra using card games. Future research can focus on analyzing the computational complexity of the algorithms used in the paper.
The paper identifies a need for a deeper analysis of the mathematical structures underlying card games.
The study is limited to the range of the observations. The paper notes that the results may not be generalizable to all sequences of rational numbers. The use of numerical experiments may introduce errors or biases.
The paper suggests that future research should focus on the development of new mathematical techniques and tools for studying the spectra of matrices associated with rational numbers. The study of the coherence phenomenon in the spectra of matrices should be continued. The use of the Anthropic AI model Claude should be further explored.
There is a gap in our understanding of adelic distributions of algebraic numbers. The paper aims to fill this gap by providing a framework for understanding adelic distributions.
The paper notes that the analogous problem over the Deligne-Mumford compactification is trivially easy.
To extend the algorithm to the case where the hypersurface defined by the vanishing of the polynomial is not smooth. To improve the efficiency of the algorithm. To apply the algorithm to various applications.
Computing points in connected components defined by a real inequation: algorithms, complexity and implementations, Part I · 2026The paper does not establish human peer review or historical priority. The paper does not characterize all of ΩF or prove a lower bound on the degree of any such example. The paper does not address a distinguished transcendental or arithmetic family.
To apply the Volterra signature to specific real-world problems. To compare the proposed algorithms with existing methods. To extend the Volterra signature to other fields such as machine learning and data analysis.
Computational aspects of the Volterra Signature · 2026The paper identifies a gap in the understanding of Desarguesian embeddability and the Pappian commutativity constraint in higher-rank projective varieties. The study aims to fill this gap by establishing a novel connection between these geometric axioms and universal incidence syzygies.
Universal Incidence Syzygies: Bridging Desarguesian Embeddability and the Pappian Commutativity Constraint in Higher-Rank Projective Varieties · 2026 · DOIFurther optimization of the algorithm. Application of the algorithm to other areas of algebraic geometry. Exploration of the use of modular and parallel strategies.
The paper identifies a gap in the understanding of T-hypersurfaces and their properties. It also identifies a gap in the understanding of the Renaudineau-Shaw spectral sequence.
Most-cited papers in Polynomial and algebraic computation
- Multilinear Extensions of Games · Management Science · 1972 · 405 citations
- Integral reduction with Kira 2.0 and finite field methods · Computer Physics Communications · 2021 · 306 citations
- Concerning Certain Linear Transformation Apparatus of Cryptography · American Mathematical Monthly · 1931 · 72 citations
- Argumentation and algebraic proof · ZDM · 2008 · 55 citations
- Polynomiography: From the Fundamental Theorem of Algebra to Art · Leonardo · 2005 · 38 citations
- Principal Landau determinants · Computer Physics Communications · 2024 · 37 citations
- Algebraic Aspects of Cut Elimination · Studia Logica · 2004 · 31 citations
- Taking Multiplicity Seriously · Contemporary Psychoanalysis · 1996 · 24 citations
- THE SCHUR AND SAMUELSON CONDITIONS FOR A CUBIC EQUATION* · Manchester School · 1990 · 24 citations
- Light Affine Set Theory: A Naive Set Theory of Polynomial Time · Studia Logica · 2004 · 24 citations
Most recent work
- Random Signs into Matchings: A Godsil-Gutman Identity, Formalized in Lean 4 (Part I) · Open MIND · 2026
- Proposing and solving olympiad geometry with guided tree search · Nature Machine Intelligence · 2026
- Leaflet Modification for Redo-TAVR · JACC: Cardiovascular Interventions · 2026
- Modular algorithms for computing Gröbner bases in free algebras · Journal of Symbolic Computation · 2026
- Algebraic independence of an Airy function, its derivative, and antiderivative · Elemente der Mathematik · 2026
- Projecting dynamical systems via a support bound · Mathematics of Computation · 2026
- S-matrices in the holomorphic modular bootstrap approach · Physics Letters B · 2026
- The ring of differential operators on a monomial curve is a Hopf algebroid · Journal of Algebra · 2026
- Graded Lawson–Stone duality · Bulletin of the London Mathematical Society · 2026
- Balancing properties of tropical moduli maps · Bulletin of the London Mathematical Society · 2026
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