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/).

The R-hulloid, in the Euclidean space R3, of the set of vertices V of a tetrahedron T is the minimal closed set containing V such that its complement is the union of open balls of radius R. When R is greater than the circumradius of T, the boundary of the R-hulloid consists of V and possibly of four spherical subsets of well defined spheres of radius R through the vertices of T. The existence of a value R & lowast; such that these subsets collapse into a point O & lowast;, in the interior of T, is investigated; in such a case O & lowast; belongs to four spheres of radius R & lowast;, each one through three vertices of T and not containing the fourth one. As a consequence, the range of rho such that Vis a rho-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

We consider the problem of certifying the feasibility of a (possibly degenerate) semidefinite programming problem represented by rational input data together with a numerical solution that is guaranteed to be sufficiently close to an exact solution of maximal rank. Our method constructs a polynomial system whose set of real solutions has an isolated point that satisfies the input program. An experimental comparison between the corresponding symbolic-numerical algorithm and the purely symbolic algorithm from [3, 5] shows that the hybrid algorithm is able to certify instances where the exact method fails.

We consider linear matrix inequalities (LMIs) A = A(0) + x(1)A(1) + center dot center dot center dot +x(n)A(n) >= 0 with the A(i)'s being m x m symmetric matrices, with entries in a ring R. When R = R, the feasibility problem consists in deciding whether the x(i)'s can be instantiated to obtain a positive semi-definite matrix. When R = Q[y(1),...,.y(t)], the problem asks for a formula on the parameters y(1),...,.y(t), 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, under genericity assumptions. The complexity of this algorithm is exponential in n, m and t but becomes polynomial in n when m and t are fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods, which are both known to be equivalent to the feasibility of certain parametric LMIs, hence demonstrating its practical interest.

This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical solvers that work with rational numbers can only find an approximate solution. We study the following question: is it possible to certify feasibility of a given SDP using an approximate solution that is sufficiently close to some exact solution? Existing approaches make the assumption that there exist rational feasible solutions (and use techniques such as rounding and lattice reduction algorithms). We propose an alternative approach that does not need this assumption. More specifically, we show how to construct a system of polynomial equations whose set of real solutions is guaranteed to have an isolated correct solution (assuming that the target exact solution is maximum-rank). This allows, in particular, to use algorithms from real algebraic geometry for solving systems of polynomial equations, yielding a hybrid (or symbolic-numerical) method for SDPs. We experimentally compare it with a pure symbolic method; the hybrid method was able to certify feasibility of many SDP instances on which the exact method failed. Our approach may have further applications, such as refining an approximate solution using methods of numerical algebraic geometry for systems of polynomial equations.

A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron $T \subset \mathbb{R}^3$ is introduced to represent the configurations of four spheres of radius $R^*$, which intersect in one point, each sphere containing three vertices of $T$ but not the fourth one. This problem is related to that of computing the largest value $r$ for which the set of vertices of $T$ is an $r$-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius $R^*$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^*$ is unique, in the second one three values $R^*$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $r$-bodies.

The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure on a given set K of a finite-dimensional Euclidean space. This problem has plenty of applications e.g. in optimization, control theory and statistics. When K is a compact semialgebraic set, the duality between the cone of moments of non-negative measures on K and the cone of non-negative polynomials on K yields an alternative: either y is a moment vector, or y is not a moment vector, in which case there exists a polynomial strictly positive on K making a linear functional depending on y vanish. Such a polynomial is an algebraic certificate of moment unrepresentability. We study the complexity of computing such a certificate using computer algebra algorithms.

Background:The Regional Basis of Solid Tumor (RBST), a clinical data warehouse, centralizes information related to cancer patient care in 5 health establishments in 2 French departments.Purpose:To develop algorithms matching heterogeneous data to "real" patients and "real" tumors with respect to patient identification (PI) and tumor identification (TI).Methods:A graph database programed in java Neo4j was used to build the RBST with data from ~20 000 patients. The PI algorithm using the Levenshtein distance was based on the regulatory criteria identifying a patient. A TI algorithm was built on 6 characteristics: tumor location and laterality, date of diagnosis, histology, primary and metastatic status. Given the heterogeneous nature and semantics of the collected data, the creation of repositories (organ, synonym, and histology repositories) was required. The TI algorithm used the Dice coefficient to match tumors.Results:Patients matched if there was complete agreement of the given name, surname, sex, and date/month/year of birth. These parameters were assigned weights of 28%, 28%, 21%, and 23% (with 18% for year, 2.5% for month, and 2.5% for day), respectively. The algorithm had a sensitivity of 99.69% (95% confidence interval [CI] [98.89%, 99.96%]) and a specificity of 100% (95% CI [99.72%, 100%]). The TI algorithm used repositories, weights were assigned to the diagnosis date and associated organ (37.5% and 37.5%, respectively), laterality (16%) histology (5%), and metastatic status (4%). This algorithm had a sensitivity of 71% (95% CI [62.68%, 78.25%]) and a specificity of 100% (95% CI [94.31%, 100%]).Conclusion:The RBST encompasses 2 quality controls: PI and TI. It facilitates the implementation of transversal structuring and assessments of the performance of the provided care.

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Padé approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fröberg’s Conjecture in commutative algebra.

A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive and efficient in the context of real algebraic geometry. Namely, given two compact disjoint basic semialgebraic sets which are contained in an $n$-dimensional box, we provide an algorithm that computes a separating polynomial greater than or equal to 1 on the first set and less than or equal to 0 on the second one.

We revisit facial reduction from the point of view of projective geometry. This leads us to a homogenization strategy in conic programming that eliminates the phenomenon of weak infeasibility. For semidefinite programs (and others), this yields infeasibility certificates that can be checked in polynomial time. Furthermore, we propose a refined type of infeasibility, which we call stably infeasible, for which rational infeasibility certificates exist and that can be distinguished from other infeasibility types by our homogenization.

Let A0, ..., An be m x m symmetric matrices with entries in Q, and let A(x) be the linear pencil A0+x1 A1 + ··· + xn An, where x=(x1,...,xn) are unknowns. The linear matrix inequality (LMI) A(x) ≥ 0 defines the subset of Rn, called spectrahedron, containing all points x such that A(x) has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice but we prove that solving such problems can be done in polynomial time if either n or m is fixed.

Let f1, ..., fm be elements in a quotient Rn/N which has finite dimension as a K-vector space, where R = K[X1, ..., Xr] and N is an R-submodule of Rn. We address the problem of computing a Gröbner basis of the module of syzygies of (f1, ..., fm), that is, of vectors (p1, ..., pm) ∈ Rm such that p1f1 + ... + pm fm = 0.An iterative algorithm for this problem was given by Marinari, Möller, and Mora (1993) using a dual representation of Rn/N as the kernel of a collection of linear functionals. Following this viewpoint, we design a divide-and-conquer algorithm, which can be interpreted as a generalization to several variables of Beckermann and Labahn's recursive approach for matrix Padé and rational interpolation problems. To highlight the interest of this method, we focus on the specific case of bivariate Padé approximation and show that it improves upon the best known complexity bounds.

We describe a protocol for generating random numbers based on the existence of quantum violations of a free Clauser-Horne-Shimony-Holt inequality, namely CHSH-3. Our method uses semidefinite programming relaxations to compute such violations. In a standard setting the CHSH-3 inequality involves two separated qutrits and compatible measurement, that is, commuting with each other, yielding the known quantum bound of $1+\sqrt{11/3} \approx 2.9149$. In our framework, $d$-dimensional quantum systems (qudits) where $d$ is not fixed a priori, and measurement operators possibly not compatible, are allowed. This loss of constraints yields a higher value for the maximum expectation of the CHSH-3 inequality. Based on such upper bound on the violation of CHSH-3, we develop a random number generator of type prepare-and-measure, but with one part. Our protocol generates a maximal entropy and its security is based, through self testing arguments, on the attainability of the maximal violation of the free CHSH-3 for quantum systems.

We consider m × s matrices (with m≥ s ) in a real affine subspace of dimension n . The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in ( [ n+m(s-r); n ]) . It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach.

This document describes our freely distributed Maple library spectra, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.

We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve.

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming (SDP). We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function.

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.