We present an algorithm that incrementally computes a Gröbner basis for the vanishing ideal of any finite set of points in an affine space under any monomial order, and we apply this algorithm to polynomial interpolation in multiple variables. For the case of distinct points, the algorithm is natural generalization of Newton’s interpolation for univariate polynomials. The time complexity in the...

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with...

The Springer variety SX is defined to be the set of flags stabilized by a nilpotent operator X. Springer varieties can be generalized to a two-parameter family of varieties called Hessenberg varieties H(X, h), defined by a nilpotent operator X and a certain step function h (or equivalently, a Dyck path). In 1992, Garsia-Procesi gave a presentation of the cohomology ring H(SX) as the polynomial ...

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of...

For any set representation (permutation representation) of the symmetric group Sn, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have t...

We show that solution sets of systems tropical differential equations can be characterised in terms monomial-freeness an initial ideal. discuss a candidate definition basis and give nonexistence result for such bases example.

Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley-Reisner ring F (G) of the broken circuit complex of G is CohenMacaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F (G) in terms of fundamental cocircuits in G. So F (G) modulo this hsop is a finite dimensional vector space. We conje...

We describe a procedure to develop a fitting basis for molecular potential energy surfaces (PESs) that is invariant with respect to permutation of like atoms. The method is based on a straightforward symmetrization of a primitive monomial basis and illustrated for several classes of molecules. A numerically efficient method to evaluate the resulting expression for the PES is also described. The...

A monomial orthogonal polynomial of several variables is of the form x−Qα(x) for a multiindex α ∈ N 0 and it has the least L2 norm among all polynomials of the form xα − P (x), where P and Qα are polynomials of degree less than the total degree of xα. We study monomial orthogonal polynomials with respect to the weight function ∏d+1 i=1 |xi| 2κi on the unit sphere Sd as well as for the related w...