Related papers: Heights of polynomials over lemniscates

We show that Mahler's classification of real numbers $\zeta$ with respect to the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence…

We prove that the expected value and median of the supremum of $L^2$ normalized random holomorphic fields of degree $n$ on $m$-dimensional K\"ahler manifolds are asymptotically of order $\sqrt{m\log n}$. This improves the prior result of…

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We…

By a generalized Delsarte polynomial we mean a Laurent polynomial whose exponent vectors are linearly independent. We consider certain monomial deformations of generalized Delsarte polynomials and study their associated differential…

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized…

We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known…

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their…

We give several characterizations of holomorphic mean Besov-Lipschitz space on the unit ball in $\cn $ and appropriate Besov-Lipschitz space and prove the equivalences between them. Equivalent norms on the mean Besov-Lipschitz space involve…

We consider the problem of computing the minimum value $f_{\min,K}$ of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can be reformulated as finding a probability measure $\nu$ on $K$ minimizing $\int_K f d\nu$.…

A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This…

We study the lemniscates of rational maps. We prove a reflection principle for the harmonic measure of rational lemniscates and we give estimates for their capacity and the capacity of their components. Also, we prove a version of Schwarz's…

Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in…

In this paper we study the average $\NL_{2\alpha}$-norm over $T$-polynomials, where $\alpha$ is a positive integer. More precisely, we present an explicit formula for the average $\NL_{2\alpha}$-norm over all the polynomials of degree…

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of…

In this paper, we develop the higher descent equations for higher gauge theories within the framework of 2-term $L_{\infty}$ algebras. Starting from a multilinear symmetric invariant polynomial, we construct a family of higher Chern-Simons…

Let l be an oriented link of d components in a homology 3-sphere. For any nonnegative integer q, let l(q) be the link of d-1 components obtained from l by performing 1/q surgery on the dth component. Then the Mahler measure of the Alexander…

We show that the set of complex points in the moduli space of polynomials of degree d corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows that for any B, the set of conjugacy…

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we…

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Assume $A$ is a basic connected and triangular algebra with $n$ pairwise non-isomorphic simple modules. We consider the {\em Coxeter transformation} $\phi_A(T)$…