Related papers: Some Notes on Flat Polynomials

A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid…

In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result…

We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational…

Let X be an analytic set defined by polynomials whose coefficients a_1,...,a_s are holomorphic functions. We formulate conditions such that for all sequences {a_(1,n)},...,{a_(s,n)} of holomorphic functions converging locally uniformly to…

We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with…

A classical theorem of Wendroff shows that one may reconstructs a sequence of orthogonal polynomials on the real line from two non-constant polynomials of consecutive degrees whose zeros strictly interlace on the real line. In this note we…

We establish necessary and sufficient conditions for an arbitrary polynomial of degree $n$, especially with only real roots, to be trivial, i.e. to have the form a(x-b)^n. To do this, we derive new properties of polynomials and their roots.…

We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree $N$ on…

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of…

We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = \eta \cdot\mathfrak o(\sqrt{-3})$, $\eta = \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map…

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in…

The spectrum of a 1+1 dimensional field theory with dynamical quarks is constructed. We focus in testing the possible brane embeddings that can support fundamental matter. The requirement on the wave function normalisation and the…

The parametrization theorem is derived in a flat nD pseudo-complex affine space. The pseudo-complex hyperbolic space accomodates n-number of uncompactified time-like extra dimensions with sugnature (s,r), where s and r are the numbers of…

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of…

We establish a pointwise convergence result for ergodic averages modeled along orbits of the form $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$, where $k$ is an arbitrary positive rational number with $\sqrt{k}\not\in\mathbb{Q}$. Namely,…

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a…

Generic Newton polygons for L-functions of exponential sums associated to Laurent polynomials in one variable are determined. The corresponding Hasse polynomials are also determined.

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative…

It is shown that Erd\"{o}s--Littlewood's polynomials are not $L^\alpha$-flat when $\alpha > 2$ is an even integer (and hence for any $\alpha \geq 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we…

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials…