Related papers: Dimension Results for Extremal-Generic Polynomial …
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous…
Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators. Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M. Any monomial ideal M can…
We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that…
The computational complexity of polynomial ideals and Gr\"obner bases has been studied since the 1980s. In recent years, the related notions of polynomial subalgebras and SAGBI bases have gained more and more attention in computational…
We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting…
The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by…
We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume…
Given a tame Galois branched cover of curves pi: X -> Y with any finite Galois group G whose representations are rational, we compute the dimension of the (generalized) Prym variety corresponding to any irreducible representation \rho of G.…
We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the…
We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to…
We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…
We introduce a toric version of the sheaf of logarithmic vector fields along a divisor of a simplicial toric variety. The notion is also relevant for algebraically independent families of polynomials in the Cox ring. We provide a…
Toric varieties are a special class of rational varieties defined by equations of the form {\it monomial = monomial}. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety $X$ contains a cover by…
We generalise the notion of Gr\"obner fan to ideals in R[[t]][x_1,...,x_n] for certain classes of coefficient rings R and give a constructive proof that the Gr\"obner fan is a rational polyhedral fan. For this we introduce the notion of…
We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a…
We study the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we prove that the polynomials representing these classes (up to suitably changing signs) are…
We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…