arXiv subjects
Publications and source records attributed to Giorgos Kotsovolis.
We show that the spectrum of every $\mathbb{R}-$isotropic homogeneous binary form $P$ of degree $n\geq3$ is an interval of the form $[0,M_P],$ where $M_P$ is some positive constant. This completes the discussion around a conjecture of Mordell from 1940 (disproved by Davenport) regarding the existence of spectral gaps for binary cubic forms and further settles Mahler's program for binary forms of every degree.
We give a complete list of the points in the spectrum $$\mathcal{Z}=\{\inf_{(x,y)\inΛ,xy\neq0}{\left\vert xy\right\vert},\,\text{$Λ$ is a unimodular rational lattice of $\mathbb{R}^2$}\}$$ above $\frac{1}{3}.$ We further show that the set of limit points of $\mathcal{Z}$ with values larger than $\frac{1}{3},$ is equal to the set $\{\frac{2m}{\sqrt{9m^2-4}+3m},\text{ where $m$ is a Markoff number}\}$.
We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of $n\times n$ matrices, the Pfaffian on the space of skew-symmetric $2n\times 2n$ matrices, and the determinant polynomial on the space of symmetric $n\times n$ matrices. In particular, let $(V,F)$ be any pair among the following: $(\textrm{Mat}_n, \det)$, $(\textrm{Skew}_{2n},\textrm{Pff})$, and $(\textrm{Sym}_n, \det).$ We then obtain an asymptotic for $$π_{V,F}(T)= \#\{v\in V: \max(|v_i|)\leq T, F(v) \text{ is prime}\},$$ that matches the Bateman-Horn prediction. The key ingredients of our proof are an asymptotic count for integral points on the level sets of $F$ given by Linnik equidistribution, a geometric approximation of the box by cones, and an upper bound sieve to bound the number of prime values missed by the approximation. In the case of the determinant polynomial on symmetric matrices, we must also use the Siegel mass formula to compute the product of local densities for the main term.
In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.
Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains Oi that extend continuously on the product of the union of Oi with Si , where Si are subsets of the boundary of Oi, open in the relative topology. The universal approximation occurs on every product of compact sets Ki such that C - Ki are connected and for some i0 it holds that Ki0 is contained in the complement of the union of Oi0 with the closure of Si0. Furthermore,we introduce some topological properties of universal Taylor series that lead to the voidance of some families of functions.