Articles

07/02/2003-- 07/02/2003

Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochner-Weitzenb\"ock formulas for gradients. As applications, we give some vanishing theorems.
Yasushi Homma
09/28/2005-- 05/20/2004

Estimating the eigenvalues on Quaternionic Kähler Manifolds

We study geometric first order differential operators on quaternionic K\"ahler manifolds. Their principal symbols are related to the enveloping algebra and Casimir elements for $\Sp(1)\Sp(n)$. This observation leads to anti-symmetry of the principal symbols and Bochner-Weitzenb\"ock formulas for operators. As an application, we estimate the first eigenvalues of them.
Yasushi Homma
08/22/2011-- 08/22/2011

Rational curves with many rational points over a finite field

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a generalization of the curve are also presented.
Satoru Fukasawa Masaaki Homma Seon Jeong Kim
08/25/2011-- 08/25/2011

A bound on the number of points of a curve in projective space over a finite field

For a nondegenerate irreducible curve $C$ of degree $d$ in ${\Bbb P}^r$ over ${\Bbb F}_q$ with $r \geq 3$, we prove that the number $N_q(C)$ of ${\Bbb F}_q$-points of $C$ satisfies the inequality $N_q(C) \leq (d-1)q +1$, which is known as Sziklai's bound if $r=2$.
Masaaki Homma
04/01/2015-- 04/01/2013

The characterization of Hermitian surfaces by the number of points

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space.
Masaaki Homma Seon Jeong Kim
07/22/2015-- 07/22/2015

Gamma Polari-Calorimetry with SOI pixels for proposals at Extreme Light Infrastructure (ELI-NP)

We introduce the concept of Gamma Polari-Calorimetry (GPC) dedicated for proposals at Extreme Light Infrastructure in the Romanian site (ELI-NP). A simulation study shows that an assembly of thin SOI pixel sensors can satisfy our requirements to GPC.
Kensuke Homma Yoshihide Nakamiya
11/08/2016-- 11/08/2016

Number of points of a nonsingular hypersurface in an odd-dimensional projective space

The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective $3$-space.
Masaaki Homma Seon Jeong Kim
09/08/2015-- 09/08/2015

The second largest number of points of plane curves over finite fields

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree $d$ over the finite field of $q$ elements is also given for $d\geq q+1$.
Masaaki Homma Seon Jeong Kim
03/05/2020-- 03/05/2020

On a Homma-Kim conjecture for nonsingular hypersurfaces

Let $X^n$ be a nonsingular hypersurface of degree $d\geq 2$ in the projective space $\mathbb{P}^{n+1}$ defined over a finite field $\mathbb{F}_q$ of $q$ elements. We prove a Homma-Kim conjecture on a upper bound about the number of $\mathbb{F}_q$-points of $X^n$ for $n=3$, and for any odd integer $n\geq 5$ and $d\leq q$.
Andrea Luigi Tironi
08/09/2016-- 09/23/2014

A bound for the number of lines lying on a nonsingular surface in $3$-space over a finite field

A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over $\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this bound is optimal for $d = 2, \sqrt{q}+1, q+1$.
Masaaki Homma Seon Jeong Kim


with thanks to arxiv.org/