Related papers: The Waring Problem of Harmonic Polynomials

Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an…

Addressing a problem posed by W. Li and A. Wei (2009), we investigate the average number of (complex) zeros of a random harmonic polynomial $p(z) + \overline{q(z)}$ sampled from the Kac ensemble, i.e., where the coefficients are independent…

We prove several new results concerning the pure quantum polynomial hierarchy (pureQPH). First, we show that QMA(2) is contained in pureQSigma2, that is, two unentangled existential provers can be simulated by competing existential and…

The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The…

This work concerns Waring decompositions of a certain kind of plane quartics of high rank. The main result is the following. Let x, l_1, ...., l_7 be linear forms and q a quadratic form on a vector space of dimension 3. If…

We study the problem of whether $\mathcal{P}_w(^nE)$, the space of $n$-homogeneous polynomials which are weakly continuous on bounded sets, is an $M$-ideal in the space of continuous $n$-homogeneous polynomials $\mathcal{P}(^nE)$. We obtain…

We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting…

Motivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of k-th powers of forms of degree…

This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…

The paper deals with the computation of the rank and the identifiability of a specific ternary form. Often, one knows some short Waring decomposition of a given form, and the problem is to determine whether the decomposition is minimal and…

The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam…

We analyze wave functions constructed as a sum of product of two-electron functions, or as a polynomial of geminals, to investigate their ability to represent the ground state of a strongly correlated few-body system. The known difficulty…

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal…

Waring's classical problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for nonabelian groups, and simple groups in particular. Here the k-th…

Let $\M$ be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into $\M$ with prescribed singularities along a…

In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics…

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop…

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…

We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…