相关论文: Parametric geometry of numbers over a number field…

We formulate a general problem: given projective schemes $\mathbb{Y}$ and $\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from $\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in $\mathbb{X}(K)$ of height at…

In a series of recent papers, W. M. Schmidt and L. Summerer developed a new theory by which they recover all major generic inequalities relating exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and find new ones. Given a…

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the…

Let $\lambda \in \mathbb{Q}\setminus \{0, -1\}$ and $l \geq 2$. Denote by $C_{l,\lambda}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=(x-1)(x^2+\lambda).$$ In this paper we give a…

We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational…

The expansion of Kummer's hypergeometric function as a series of incomplete Gamma functions is discussed, for real values of the parameters and of the variable. The error performed approximating the Kummer function with a finite sum of…

Upper bounds of the Hausdorff volume of scalar gradient field graphs are derived by means of geometric measure theory. The approach reproduces that scalar gradient fields along a mean imposed scalar gradient become space filling for…

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we…

The well known theorems of Khintchine and Jarn\'ik in metric Diophantine approximation provide comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various…

Quantum geometry is a differential geometry based on quantum mechanics. It is related to various transport and optical properties in condensed matter physics. The Zeeman quantum geometry is a generalization of quantum geometry including the…

Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we show that German's transference inequalities between the two most classical exponents of uniform Diophantine…

This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and…

The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…

We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We show that the perturbation expansion of…

We demonstrate how one can see quantization of geometry, and quantum algebraic structure in supersymmetric gauge theory.

We show that the sum over geometries in the Lorentzian 4-D state sum model for quantum GR in [1] includes terms which correspond to geometries on manifolds with conical singularities. Natural approximations suggest that they can be…

General Relativity describes gravity in geometrical terms. This suggests that quantizing such theory is the same as quantizing geometry. The subject can therefore be called quantum geometry and one may think that mathematicians are…

I have shown that the field defined by the Wheeler-DeWitt equation for \textit{pure gravity} is neither a standard gravitational field nor the field representing a particular universe. The theory offers a unified description of geometry and…

We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and apply them with a 2-descent argument to obtain bounds for families of elliptic curves over certain imaginary quadratic number fields with class number 1. One such…

We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…