Open research questions in Polynomial and algebraic computation
52 unresolved questions extracted from the limitations and future-work sections of 216 Polynomial and algebraic computation papers in our library. Each links back to the study that raised it.
What the literature leaves open
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.
To prove the result in all characteristics. To apply the method to other problems in algebraic geometry. To further explore the use of AI agents in mathematics.
The paper identifies a gap in the understanding of the Hodge bundle. The paper notes that the analogous problem over the Deligne-Mumford compactification is trivially easy.
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.
Further study of the quaternionic generalization of the Eneström-Kakeya Theorem. Exploration of the applications of the results in physical systems and other fields.
The quaternionic context has not been fully explored. The distribution of zeros for polynomials with quaternionic variables and quaternionic coefficients has not been well covered in the literature.
The paper notes that the combinatorial complexity of the problem is significant. It identifies the need to develop efficient algorithms for solving dimension and associativity equations. The paper notes that the extension to the 1-Frobenius case is a challenge.
The paper does not generalize to fusion rings. The proposition in [24, Theorem 1.6 and Proposition 4.5(iv)] does not extend to fusion rings. The extension to the 1-Frobenius case remains an open question.
Future research can focus on the application of the results to other association schemes. Future research can explore the properties of regularized bivariate affine q-Krawtchouk polynomials.
The paper identifies a gap in the representation of the first eigenmatrix of a translation association scheme on Matd×n(GR(p2, r)).
The lack of an exact algebraic characterization of the optimal dispersion distance d₁₀. The need for certification of the irreducibility of the polynomial P₁₈ over ℚ.
Exact Minimal Polynomial and Algebraic Degree of the Optimal 10-Point Packing Distance in a Unit Square · 2026 · DOIThe paper identifies a gap in the literature regarding the algebraic certification of the minimal polynomial of the optimal 10-point packing distance. The paper identifies a gap in the literature regarding the determination of the Galois group of the minimal polynomial.
An Independent Symbolic Certification and Galois Group Proof for the 10-Point Square Packing Distance · 2026 · DOIThe results are limited to the case of homogeneous ideals in polynomial rings. The paper does not provide a complete characterization of the relationships between Hilbert functions and Hermitian polynomials.
To study the relationships between Hilbert functions and Hermitian polynomials in more general contexts. To apply the results to the study of proper holomorphic mappings between complex unit balls. To further develop the theory of Macaulay's Theorem and its applications.
Further research can be done to apply the results to other areas. The generalization can be used to study polynomials vanishing on certain sets.
The existing results on Nullstellensätze have limitations. The paper addresses these limitations by providing a generalization.
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.
The 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 · DOIThe 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 · DOIThe 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 · DOIFuture research can focus on extending the result to other models of computation. Future research can focus on improving the efficiency of algorithms for computing factors of polynomials. Future research can focus on applying the result to other areas of computer science.
The gap in current research is the study of other natural models of computation for polynomials. The gap is the lack of a complete characterization of the factors of polynomials with small circuits or formulas.
Further optimization of the algorithm. Application of the algorithm to other areas of algebraic geometry. Exploration of the use of modular and parallel strategies.
Most-cited papers in Polynomial and algebraic computation
- Random Signs into Matchings: A Godsil-Gutman Identity, Formalized in Lean 4 (Part I) · Open MIND · 2026 · 16 citations
- Parallel Digraphs-building Computer Algorithm for Finding a Set of Characteristic Polynomial Realisations of Dynamic System · Journal of Automation Mobile Robotics & Intelligent Systems · 2016 · 5 citations
- Leaflet Modification for Redo-TAVR · JACC: Cardiovascular Interventions · 2026 · 3 citations
- DYNAMICS OF A FAMILY OF RATIONAL OPERATORS OF ARBITRARY DEGREE · Mathematical Modelling and Analysis · 2021 · 2 citations
- An Alternative Equation for Generalized Polynomials of Degree Two · Annales Mathematicae Silesianae · 2023 · 1 citations
- Morgan-Voyce Polynomial Approach for Quaternionic Space Curves of Constant Width · Foundations of Computing and Decision Sciences · 2021 · 1 citations
- An Iterative Algorithm for Determining the Greatest Common Divisor of Two or More Univariate Polynomials · Mathematics and Informatics · 2024 · 1 citations
- An Accessible Proof of Hurwitz’s Sums of Squares Theorem · Mathematics Magazine · 2022 · 1 citations
- Cubic Equations Through the Looking Glass of Sylvester · College Mathematics Journal · 2022 · 1 citations
- Analogy and Generalization as a Driving Force of Learning Mathematics – the Case of a Matrix Analog of a Zero of a Polynomial · PRIMUS · 2024 · 1 citations
Most recent work
- Random Signs into Matchings: A Godsil-Gutman Identity, Formalized in Lean 4 (Part I) · Open MIND · 2026
- Leaflet Modification for Redo-TAVR · JACC: Cardiovascular Interventions · 2026
- Integer Polynomials Expressible as Sums of Cubes · Mediterranean Journal of Mathematics · 2026
- The Diogo Sousa Algorithm: An Exact Triadic Identity Between Binary and Decimal Bases · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Specialization and rigidity · Journal of Algebra · 2026
- Constructing Geometric Realizations of Birational Maps Between Mori Dream Spaces · Experimental Mathematics · 2026
- On the extension of a class of Hermite multivariate interpolation problems · Numerical Algorithms · 2026
- CNRS Arch: A Combined Architecture for Complex Numeric Representation Systems: Arithmetic, Analytic Continuation, Operator Calculus, and Physical Structure · Zenodo (CERN European Organization for Nuclear Research) · 2026
- ** Resolution to P=NP Proven True ** and it's Mathematics Construct For **Polynomial Time Computations ** · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Boost.Geometry: A Generic C++ Geometry Library · Journal of Open Source Software · 2026
Find a gap in your own Polynomial and algebraic computation sub-topic
This page shows what the Polynomial and algebraic computation literature already flags as unresolved. To narrow it to your specific question, run the guided finder — it searches the gap library on demand and checks candidates against 250M+ OpenAlex works.
Open the Research Gap Finder →