Related papers: On an arithmetic inequality on $\mathbb{P}^1_{\mat…
We prove that the previously established inequality of different metrics for algebraic polynomials is sharp in the sense of order.
For a divisor $D$ on a tropical variety $X$, we define two amounts in order to estimate the value of $h^{0}(X,D)$, which are described by terms of global sections and computed more easily than $h^{0}(X,D)$. As an application of its…
The volume of a Cartier divisor on a projective variety is a nonnegative real number that measures the asymptotic growth of sections of multiples of the divisor. It is known that the set of these numbers is countable and has the structure…
For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous…
We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov…
In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement.
We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the…
The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to…
We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the…
Given an endomorphism f of projective space, we exhibit explicit bounds on the difference between the naive height of a divisor and its canonical height relative to f.
In this paper, given a convex, bounded, open set $\Omega \subset \mathbb{R}^n$ we prove a sharp inequality involving the Laplacian torsional rigidity and both the perimeter and the measure of the domain. Our result generalizes to arbitrary…
In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that…
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented
In the setting of Arakelov geometry over adelic curves, we introduce the $\chi$-volume function and show some general properties. This article is dedicated to talk about the continuity of $\chi$-volume function. By discussing its…
We use an injection method to prove a new class of partition inequalities involving certain $q$-products with two to four finitization parameters. Our new theorems are a substantial generalization of work by Andrews and of previous work by…
In this paper, inspired by Schur's comparison theorem about curves in Euclidean space, we mainly provide a Schur's type volume comparison theorem, which is about the volumes of the boundaries of open balls in a complete $n$-dimensional…
In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in n-dimensions. Moreover, we discuss how the latter are related to the monotonicity of an…
We study the multi-height distribution of rational points of smooth, projective and split toric varieties over $\mathbf{Q}$ using the lift of the number of points to universal torsors.
Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, with smooth generic fiber $X_K$. We give a formula that relates the dimension of the first Arakelov-Chow vector space of $X$ with the Mordell-Weil rank of the…
We offer new proofs, refinements as well as new results related to classical means of two variables, including the identric and logarithmic means.