Related papers: Newton numbers, vanishing polytopes and algebraic …
Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values…
We study unbounded 2-dimensional metric polytopes such as those arising as K\"ahler quotients of complete K\"ahler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete…
We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study…
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart…
For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving mixed…
Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular…
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree…
The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial…
We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a…
We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and…
In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as…
We use a polynomial decomposition result by Stapledon to show that the numerator polynomial of the Ehrhart series of an open polytope is the difference of two symmetric polynomials with nonnegative integer coefficients. We obtain a related…
We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…
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…
We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit…
A semi-classical check of the Goddard-Nuyts-Olive (GNO) generalized duality conjecture for gauge theories with adjoint Higgs fields is performed for the case where the unbroken gauge group is non-abelian. The monopole solutions of the…
We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are…
A theorem due to D. Bernstein states that Euler characteristic of a hypersurface defined by a polynomial f in (C\{0})^n is equal (upto a sign) to n! times volume of the Newton polyhedron of f. This result is related to algebaric torus…
Let $\chi$ be an order $c$ multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and $T$-adic Newton polygons of $f$. When $p>(d-e)(2d-1)$, we give a lower bound of…
A set $\mathcal{S}$ of points in $\mathbb{R}^n$ is called a rationally parameterisable hypersurface if $\mathcal{S}=\{\boldsymbol{\sigma}(\mathbf{t}): \mathbf{t} \in D\}$, where $\boldsymbol{\sigma}: \mathbb{R}^{n-1} \rightarrow…