Related papers: A note on free determinantal hypersurface arrangem…
We give an explicit presentation of the module of differentials of order $n$ of a finitely generated algebra via a higher-order Jacobian matrix. We use the presentation to study some aspects of this module in the case of hypersurfaces. More…
In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite…
In the present note we construct new families of free plane curves starting from a curve $C$ and adding high order inflectional tangent lines of $C$, lines joining the singularities of the curve $C$, or lines in the tangent cone of some…
We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w…
In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via…
We study the structure of the algebra of polynomial invariants for the usual conjugation action of the complex special, SO_n, and general, O_n, orthogonal group on the space of traceless n by n complex matrices. (Note that these two…
In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a…
We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple…
We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman, and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices.…
We construct families of hyperbolic hypersurfaces $X_d\subset\mathbb{P}^{n+1}(\mathbb{C})$ of degree $d\geq {\textstyle{(\frac{n+3}{2})^2}}$.
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…
We consider a particular type of matrices which belong at the same time to the class of Hessenberg and Toeplitz matrices, and whose determinants are equal to the number of a type of compositions of natural numbers. We prove a formula in…
In this paper, we establish the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,\alpha}$ fixed graphical boundaries under a barrier condition. This result generalizes…
The method of imsets, introduced by Studen\'y, provides a geometric and combinatorial description of conditional independence statements. Elementary conditional independence statements over a finite set of discrete random variables…
We construct an extension of Gaussian elimination to show that if $\mathbb{F}$ is a topological field, then there is a transitive, free, and continuous action of a natural quotient of $GL_k(\mathbb{F}) \times GL_{k+1}(\mathbb{F})$ on the…
In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose…
One studies certain degenerations of the generic square matrix over a field $k$ along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these…
We compute small rational models for configuration spaces of points on oriented surfaces, as right modules over the framed little disks operad. We do this by splitting these surfaces in unions of several handles. We first describe rational…
We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated…
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function…