Related papers: Exact matrix formula for the unmixed resultant in …
We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other…
We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [HM15]. We show that the algebraicity of such a…
We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gr\"obner bases. Each variety encodes a collection of matroids, whose flats…
Consider a regular triangulation of the convex-hull $P$ of a set $\mathcal A$ of $n$ points in $\mathbb R^d$, and a real matrix $C$ of size $d \times n$. A version of Viro's method allows to construct from these data an unmixed polynomial…
For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of…
We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial formulas for the dimensions of their associated subvarieties under genericity…
An exact invariant is derived for three-dimensional Hamiltonian systems of $N$ particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function $f_{2}(t)$, embodying a solution…
We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…
We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps…
Algorithms for Gaussian process, marginal likelihood methods or restricted maximum likelihood methods often require derivatives of log determinant terms. These log determinants are usually parametric with variance parameters of the…
The finite difference equation system introduced by Christiane Poupard in the study of tangent trees is reinterpreted in the alternating permutation environment. It makes it possible to make a joint study of both tangent and secant trees…
We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the…
We derive an exact, analytic expression for the fourth virial coefficient of a system of polydisperse spheres under the constraint that the smallest sphere has a radius smaller than a given function of the radii of the three remaining…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed…
We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{\gamma+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard…
This work presents closed formulas for determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.
We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of perfect Lie groups with a polynomial error term. Even for semisimple quotients, our result provides the first infinite…