Related papers: Dwork hypersurfaces of degree six and Greene's hyp…

In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^3$ whose complements is…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of…

We establish an asymptotic formula for the number of points of bounded height on a singular hypersurface of the triprojective space. We will see that the final result is in accordance with Batyrev-Manin conjecture. The method used is a…

We use the Dwork-Frobenius operator to prove an integrality result for $A$-hypergeometric series whose coefficients are factorial ratios. As a special case, we generalize one direction of a classical result of Landau on the integrality of…

In this note we study common preperiodic points of rational maps of the Riemann Sphere. We show that given any degrees $d_1,d_2\geq2$, outside a Zariski closed subset of the space of pairs of rational maps $(f,g)$ of degree $d_1$ and $d_2$…

For $n\geq 3$, let $\mathscr{M} \subseteq\mathbb{R}^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d\in \mathbb{R}_{>1}$, with non-vanishing curvature away from the origin. Consider the number…

We derive explicit representation formulae of Green functions for GJMS operators on $n$-spheres, including the fractional ones. These formulae have natural geometric interpretations concerning the extrinsic geometry of the round sphere.…

We give defining equations for function fields over finite fields with many rational places. They are obtained from composita of quadratic extensions of the rational function field.

Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified…

In this paper, we give an overview of the various general methods in computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo $p^m$ of the zeta function of a…

We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted…

In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of $\mathbb{P}^{n}$ of fixed degree. Our strategy is based on the product formula established by Bright,…

We study degree two unirational parameterizations of geometrically rational surfaces over the real field.

This paper examines the effectiveness of the Dyson-Schwinger (DS) equations as a calculational tool in quantum field theory. The DS equations are an infinite sequence of coupled equations that are satisfied exactly by the connected Green's…

The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map $f:{\Bbb C} \to X$ to…

Creative telescoping algorithms compute linear differential equations satisfied by multiple integrals with parameters. We describe a precise and elementary algorithmic version of the Griffiths-Dwork method for the creative telescoping of…

We introduce fourth fundamental form $IV,$ and $i$-th curvature formulas of hypersurfaces in the four dimensional Euclidean geometry ${\mathbb{E}}^{4}$. Defining fourth fundamental form and $i$-th curvatures for hypersurfaces, we calculate…

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…