The Relative Eigenvector Problem (REP) generalizes the classical eigenvector problem and arises naturally in computational group theory and representation theory. It plays a central role in methods for constructing finite simple subgroups of exceptional algebraic groups, as well as in algorithms for computing tensor decompositions and systems of imprimitivity of group representations.In this paper, we investigate both the computational complexity and the practical tractability of REP. We show that a decision variant of REP is NP-complete, over fixed finite fields, via a direct reduction from CNF-SAT. We then relate REP to the MinRank problem with target rank 1 and use algebraic techniques involving determinantal ideals to derive complexity bounds and identify broad parameter regimes in which generic instances of REP can be solved efficiently.We conclude with an application to the classification of embeddings of the alternating group A6 into algebraic groups of type F4. This example demonstrates how REP-based methods may admit spurious solutions and how additional polynomial constraints can be used to refine REP instances to recover genuine embeddings.
This paper aims to provide an algorithmic method for analyzing Jacobi stability of systems of second order ordinary differential equations (ODEs) using Kosambi-Cartan-Chern (KCC) theory. We develop an efficient symbolic program using Maple for computing the second KCC invariant for systems of second order ODEs in arbitrary dimension. The program allows us to systematically analyze Jacobi stability of a system of second order ODEs by means of real solving and solution classification using symbolic computation. Moreover, we present an alternative method for detecting Jacobi stability for systems of second order ODEs by making use of parametric discriminants for multiplicities of univariate polynomials. This alternative method is based on the expression of the deviation equation, which appears to be useful for detecting Jacobi stability of non-parametric models. The effectiveness of the proposed approach is illustrated by a model of wound strings, a two-dimensional airfoil model with cubic nonlinearity in supersonic flow and a 3-DOF tractor seat-operator model. The computational results on Jacobi stability of these models are further verified by numerical simulations. Our algorithmic approach allows us to detect hand-guided computation errors in published papers. In ad- dition, we conduct a comparative analysis of the Jacobi stability and the linear stability of these models. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Legendre pairs are fundamental blocks for constructing Hadamard matrices, i.e. n×n matrices with elements in {1, -1} whose inverse is the transpose scaled by 1/n. The most important conjecture relating to Hadamard matrices is their existence when n is a multiple of 4. Although algebraic constructions have been proposed for some specific values, no general construction for arbitrary n has been described and they are usually found computationally for a given n. Legendre pairs have proven to be a very valuable tool in validating the conjecture up to a point. However, a brute-force search for these pairs is infeasible even for moderate values of n. In this work, we propose to reduce the computational search by improving on a previous technique that fixes the values of partial sums, also known as M-compression, of the Legendre pairs of length pq2. We propose to study other techniques i.e. decimations, cosets, and supplementary different sets (SDS) to obtain further improvements. Our initial implementation validates this approach, as it allows us to find solutions with a common structure for LP(63) and LP(75). Future work includes extending the search for greater lengths to verify the existence of such pairs. We remark that the smallest open case for a Hadamard matrix of order n is n=668, which implies solving this conjecture up to LP(333).
Bavula proved in Bavula (2013) that the ring 1[1(k) of polynomial ordinary integro-differential operators over a field k of characteristic zero is coherent in the sense that the left/right kernel of every finite rectangular matrix with entries in I[1(k) is a finitely generated left/right 1[1(k)-module. Unfortunately, Bavula's proof is not algorithmic. This paper gives an algorithmic proof of the coherence of 1[1(k). We show that the kernel computation can be reduced to a kernel computation in the ring of skew Laurent polynomials I1(k) = k[H] congruent to I-1[(k)/(e)-where (e) is the only two-sided ideal of 1[1(k) defined by the evaluation operators-and the computation of the polynomial solutions of linear polynomial integro-differential systems. These two problems are shown to be effective. We also prove that 1[1(k) is an effective Cramer ring (i.e., a computable ring Barakat and Lange-Hegermann (2011); Posur (2021) or a coherent strongly discrete ring Mines et al. (1988)) in the sense that the solutions of every linear system of the form AX = Y or X A = Y, where A and Y are two fixed rectangular matrices with entries in 1[1(k), can be parametrized effectively. The algorithmic proof of the coherence (respectively, the Cramer property) of 1[1(k) allows us to develop an algorithmic elimination theory (respectively, an effective homological algebra) for linear systems of polynomial integro-differential equations with separable polynomial kernels. Finally, the algorithms given here are implemented in a MAPLE package called Bavula Cluzeau et al. (2025c). (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Today, symbolic mathematical computation is taken for granted as part of the scientific infrastructure, but it has not always been so. This paper provides a historical survey of the discipline’s formative decade, 1965–1975, viewed from a 50 year perspective. This span of years saw the evolution from a few specialized programs with naive algorithms to integrated systems with substantial capabilities. We highlight some of the important early figures in the field and the innovations upon which the current generation of systems and algorithms are built. By revisiting a period unfamiliar to most current readers, this survey aims to shed light on once-pressing issues that are now largely resolved and to highlight how some of today’s challenges were recognized earlier than expected.
The determinantal ideal It is the ideal generated by all the tminors of a generic matrix. In this paper, we study the minimal generators of the initial ideal in(It1 & centerdot; & centerdot; & centerdot; Itr) of the product of determinantal ideals: theoretically no general explicit description of these minimal generators is known and computationally a Gr & ouml;bner basis of It1 & centerdot; & centerdot; & centerdot; Itr is typically intractable. The degree bounds on the minimal generators of in(It1 & centerdot; & centerdot; & centerdot; Itr), and thus on the polynomials in the reduced Gr & ouml;bner basis of It1 & centerdot; & centerdot; & centerdot; Itr, are first provided. Specifically, we show that an upper bound on the degrees of the minimal generators is t1 + & sum;rk=2 k(tk-1) by analyzing the divisibility between monomials in the initial ideal via special functions applied to increasing decompositions of the monomials. For the special case of in(IaIb), we prove the existence of minimal generators of each degree between a + band the presented upper bound and thus fully characterize the degrees of its minimal generators. Using these degree bounds, we formulate an optimized version of Buchberger algorithm adapted to these products of determinantal ideals. We then investigate the algebraic and combinatorial properties of the minimal generators of in(It1 & centerdot; & centerdot; & centerdot; Itr). By analyzing the length of the longest weakly decreasing subsequence of a monomial, we present several necessary or sufficient conditions for it to be a minimal generator. In the special case of in(Itk1 1Ik2 t2 ) with t1 = t2 +1, these conditions lead to a complete characterization of its minimal generators via the path diagram we introduce and hence an explicit counting formula for the number of minimal generators. Finally, using Knuth relations, we derive explicit rules for constructing new minimal generators from a given collection of minimal generators of in(It1 & centerdot; & centerdot; & centerdot; Itr ). (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of Rn. In this framework, a CAD 'is adapted to a given set S if S is a union of cells of '. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. We thus consider the set CADr(F) of CADs of class Cr (r is an element of N boolean OR{infinity, omega}) that are adapted to a finite family F of semi-algebraic sets of Rn, endowed with the refinement partial order and we study the existence of minimal and minimum element in CADr(F). We show that for every such F and every ' is an element of CADr(F), there is a minimal CAD of class Cr adapted to F and smaller (i.e. coarser) than or equal to '. In dimension n = 1 or n = 2, this result is strengthened by proving the existence of a minimum element in CADr(F). In contrast, for any n >= 3, we provide explicit examples of semi-algebraic sets whose associated poset of adapted CADs of class Cr does not admit a minimum. We then introduce a reduction relation on CADr(F) in order to define a theoretical algorithm for the computation of minimal CADs and we characterise those semi-algebraic sets F for which CADr(F) has a minimum by means of confluence of the associated reduction system. We finally provide practical criteria for deciding if a semi-algebraic set does admit a minimum CADr and apply them to describe various concrete examples of semi-algebraic sets, along with their minimum CAD of class Cr. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing algorithms that compute such representations as a linear relation between the iterates of an elementary operator known as a pseudo-linear map. Algorithms of this form have been designed and used for solving various computational problems, in different contexts, including effective closure properties for linear differential or recurrence equations, the computation of a differential equation satisfied by an algebraic function, and many others. We propose a unified approach for establishing precise degree bounds on the solutions of all these problems. This approach relies on a common structure shared by all the specific instances of the class. For each problem, the obtained bound is tight. It either improves or recovers the previous best known bound that was derived by ad hoc methods.
Suppose that f (x) E 1[8[x1, ... , xn] and g(x) E 1[8[x1, ... , xn] are two real polynomials of degree d in n variables. If the polynomials f and g are the same up to orthogonal symmetry, a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element R E O(n) such that f (R inverted perpendicular x) = g(x). One may directly solve this problem by constructing a nonlinear system of equations induced by the relation f (R inverted perpendicular x) = g(x) along with the identities of the orthogonal group. However, this approach becomes quite computationally expensive for larger values of nand d. To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analys is (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the R E O(n) such that f (R inverted perpendicular x) = g(x). (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
MacMahon introduced partition analysis in his book “Combinatory Analysis” as a computational technique for solving problems related to systems of linear Diophantine equations and inequalities. This paper aims to develop a fundamental computational framework for MacMahon's partition analysis. As applications, we present simplified computations for “Han's formula”, the “k-gon partitions problem”, and the “two-dimensional problem”. Moreover, we apply our method to solve a challenging problem.
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and fast-to-compute invariant known as the persistence diagram. However, this is no longer the case for maps between persistence modules (i.e. persistence maps). We propose a new invariant for persistence maps, consisting of a partial matching between the persistence diagrams of the domain and codomain modules. We show that this invariant is additive with respect to the direct sum decomposition of persistence maps, is more discriminative than the image module, and is computable in cubic time. Furthermore, we provide an implementation and demonstrate its efficiency by integrating it with edge collapse techniques for flag complexes (e.g., Vietoris-Rips complexes). As a key technical contribution, we describe how to induce a persistence map between two flag complexes that have been independently simplified via edge collapses, even when a direct simplicial map between them is no longer available. (c) 2026 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
We consider linear matrices A = A0 + x1 A1 + & centerdot;& centerdot; & centerdot;+ xnAn,where the Ai's are m & times; m symmetric matrices with entries in a ring 7Z. When 7Z = & Ropf;, the feasibility problem consists in deciding whether the linear matrix inequality (LMI) A >= 0 is feasible, i.e., the xi's can be instantiated to obtain a positive semi-definite matrix. When 7Z = & Qopf;[y1, ... , yt], the problem asks for a formula on the parameters y1,...,yt, which describes the values of the parameters for which the specialized LMI is feasible. This problem can be solved using general quantifier elimination algorithms, with a complexity that is exponential in n. In this work, we leverage the LMI structure of the problem to design an algorithm that computes a formula Phi describing a dense subset of the feasible region of parameters. The complexity of this algorithm is exponential in n, m and t but becomes polynomial in n when m and tare fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods. Both problems are known to be equivalent to the feasibility of certain parametric LMIs and lead instances which are out of reach of the previous state of the art. We demonstrate that our implementations of our algorithm can tackle these problems in practice, already for small values of n, m and t. (c) 2026 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http:// creativecommons.org/licenses/by-nc/4.0/).
Many algorithms in computational algebraic geometry for understanding properties of real algebraic sets depend upon computing smooth sample points, most notably algorithms for computing the dimension of a real algebraic set. Thus, computing smooth sample points for a given real algebraic set is an important problem that can have many challenges due to singularities. An approach by the first two authors and Agnes Szanto (1966-2022) obtained smooth sample points by computing the critical points of a possibly high degree polynomial that vanishes on the singular set but does not vanish on the real algebraic set identically. This paper shows that smooth sample points can be obtained by using limits of perturbations which reduces the degree of the objective polynomial under consideration by avoiding possibly extensive deflation computations. This approach is then applied to computing the dimension of a real algebraic set. (c) 2026 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
Quaternion polynomials and their factorizations have attracted a lot of attention in recent years. This interest stems from the relationships between linear factors of such polynomials and the geometry of a mechanical linkage capable of realising the motion it parametrises. This led ultimately to an extension of Kempe's universality theorem to rational, spatial curves. Generalising these results to the multivariate case is of great interest to roboticists, as it could give a purely algebraic design method for parallel manipulators. In this article, we describe a factorization algorithm for general multivariate quaternion polynomials, as well as characterise some classes of polynomials admitting multiple non-equivalent factorizations. One of these classes-one we called "aromatic"- is particularity interesting from the perspective of theoretical kinematics, as it is factorable into polynomials parametrising rotations about constant axes. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study terminal Q-Fano 4-folds of index 1 realized as complete intersections in weighted projective space and their anticanonical linear sections. We first construct families of terminal Q-Fano 4-folds of index 1 in low-codimension weighted complete intersection models. We then investigate whether the anticanonical linear system contains a divisor that is a quasismooth Calabi-Yau 3-fold with isolated canonical singularities. Building on Qureshi (2025), which treated the cases h0(-KX) = 0 and h0(-KX) >= 2, we complete the analysis by addressing the borderline case h0(-KX) = 1. We also enlarge the list of examples in the previously studied cases by performing computations with larger search bounds. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we revisit, in a unified presentation, different techniques from polyhedral geometry that can support quantifier elimination over the integers. We propose a few improvements and report on a comparative implementation of these techniques.
Let k be a field with characteristic zero. Let f is an element of k[x, y] be a reduced homogeneous polynomial with degree d >= 1. We set I := (f) and B := k[x, y]/I. Let n >= 2 be a positive integer. In this article, we compute the module of logarithmic differential operators along I, with order n, from the datum of the polynomial f. For n = 2, we show that our method produces a closed-form presentation of this module, which yields, as a by-product, an effective proof of the Nakai conjecture for the reduced (not necessarily irreducible) homogeneous plane curve singularities. As a theoretical cornerstone of this computation, we begin by proving that the order filtration on the (left) B-module of the differential operators, with order n, denoted by Diff(k)(n)(B), actually defines a grading on Diff(k)(n)(B).
This paper studies the algorithmic decomposition of polynomial ideals using the sum-and-quotient operation, a key technique implicitly involved in the process of computing Hilbert polynomials. We extend the framework of characteristic decomposition algorithms that maintain Hilbert polynomial relationships to the positive-dimensional case. In particular, we propose algorithms that successively apply the sum-and-quotient lemma to decompose any given polynomial ideal into finitely many ideals generated by regular sets or regular sequences, such that certain relations among the zero sets and the Hilbert polynomials are simultaneously preserved. This approach offers a new framework for representing the zero sets of polynomial ideals with multiplicities and reveals inherent connections among key concepts in the algorithmic theories of triangular sets, Gr & ouml;bner bases, and Hilbert polynomials. We provide examples to illustrate computational properties and contrasts between our method and pseudo-division-based triangular decomposition algorithms. Experimental results demonstrate that the performance of our method is comparable with the existing triangular decomposition algorithms based solely on Gr & ouml;bner bases computation. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Triangular decomposition is a versatile computational tool for studying polynomial ideals, but the complexity is not well understood due to its intricate behaviors. Inspired by the recent interplay between polynomial system solving and chordal graphs, in this paper we analyze the complexity of triangular decomposition by using chordal graphs. We first introduce a new vertex ordering of graphs called the substrong elimination one which characterizes strongly chordal graphs. Using this ordering, we propose a polynomial selection strategy for triangular decomposition over F2 and show that for polynomial sets with strongly chordal associated graphs, the variables of any polynomial occurring in the decomposition with this strategy are contained in certain maximal clique. For & ell; polynomials in n variables with such an associated graph of treewidth m, under certain worst-case assumptions, the complexity of triangular decomposition over F2 using this strategy is proved to be ((m & ell;)n-1) O 4m & ell;n , smaller than the original O(& ell;n) when m << n. n-1 To extend these results to other algorithms for triangular decomposition over any field, we introduce the concept of transitive chordal graph based on the transitive vertex ordering. Then when the transitive perfect elimination ordering is used for a collection of algorithms, the similar inclusion of the variables in the maximal cliques is proved without requiring additional selection strategies. As a direct consequence, these algorithms are chordality-preserving when combined with transitive chordal graphs. At the end, we provide complexity analyses for two algorithms using transitive chordal graphs of bounded treewidths. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
