Related papers: Non-negative polynomials on generalized elliptic c…

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and…

A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In the case where $P$ has an integer root of multiplicity $1$, we show that if $A…

A generic orthotope is an orthogonal polytope whose tangent cones are described by read-once Boolean functions. The purpose of this note is to develop a theory ofEhrhart polynomials for integral generic orthotopes. The most remarkable part…

Hyperbolic polynomials elegantly encode a rich class of convex cones that includes polyhedral and spectrahedral cones. Hyperbolic polynomials are closed under taking polars and the corresponding cones, the derivative cones, yield…

We study the generalized roundness of finite metric spaces whose distance matrix $D$ has the property that every row of $D$ is a permutation of the first row. The analysis provides a way to characterize subsets of the Hamming cube $\{ 0, 1…

The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of GL(2,Z/nZ) where n is an integer. We know that, in general,…

We establish direct evidence of the arithmetic significance of plectic Stark-Heegner points for elliptic curves of arbitrarily large rank. The main contribution is a proof of the algebraicity of plectic points associated to polyquadratic CM…

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

We study real trigonal curves and elliptic surfaces of type $\I$ (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's \emph{dessins d'enfants}. We give a…

An elliptic curve $E$ over $\mathbb{Q}$ is said to be good if $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a…

We give more or less explicit equations for all two-dimensional cusp singularities of embedding dimension at least 4. They are closely related to Felix Klein's equations for universal curves with level n structure. The main technical result…

Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the…

We study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codes. We prove that if $\mathcal X$ is a superelliptic curve defined over…

We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a…

Hankel spectrahedra are the dual convex cones to the cone of sums of squares of real polynomials, and we study them from the point of view of convex algebraic geometry. We show that the Zariski closure of the union of all extreme rays of…

Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex-labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies…

We give two characterizations of cones over ellipsoids in real normed vector spaces. Let $C$ be a closed convex cone with nonempty interior such that $C$ has a bounded section of codimension $1$. We show that $C$ is a cone over an ellipsoid…

Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of…

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 consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with…