Related papers: A Gr\"obner basis for the graph of the reciprocal …

A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths.…

Applications of the H\"uckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is…

Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

Incidence-based generalizations of cycle covers, called contributors, extend the Harary-Sachs coefficient theorem for characteristic polynomials of the adjacency matrix of graphs. All minors of the Laplacian resulting from an integer matrix…

We obtain a combinatorial expression for the coefficients of the boundary map of real isotropic and odd orthogonal Grassmannians providing a natural generalization of the formulas already obtained for Lagrangian and maximal isotropic…

Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear…

We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane…

Consider a hierarchical log-linear model, given by a simplicial complex, $\Gamma$, and integer matrix $A_\Gamma$. We give a new characterization of the rank of $A_\Gamma$ given by a logarithmic transformation on the exponential Hilbert…

We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gr\"obner bases are a…

Let $I$ be an ideal of the polynomial ring $A[x]=A[x_1,...,x_n]$ over the commutative, noetherian ring $A$. Geometrically $I$ defines a family of affine schemes over $\Spec(A)$: For $\p\in\Spec(A)$, the fibre over $\p$ is the closed…

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal…

multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Nonzero multiplication operators on $L^2$ spaces of functions are never compact and then such…

It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting…

Let $G$ be 2-generated group. The generating graph $\Gamma(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a…

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer…

The relationship between a set of design points and the class of hierarchical polynomial models identifiable from the design is investigated. Saturated models are of particular interest. Necessary and sufficient conditions are derived on…

Let $U$ be either classical or quantized universal enveloping algebra of $\s\l(n+1)$ extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in $U$ over the extended Cartan subalgebra diagonalizing the…

For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…