Let $(G,\rho,X)$ be a prehomogeneous vector space with singular set $S$ over an algebraic number field $F$. The main result of this paper is a proof for the convergence of the zeta fucntions $Z(\Phi,s)$ associated with $(G,\rho,X)$ for large Re $s$ under the assumption that $S$ is a hypersurface. This condition is satisfied if $G$ is reductive and $(G,\rho,X)$ is regular. When the connected component $G_x^0$ of the stabilizer of a generic point $x$ is semisimple and the group $\Pi_x$ of connected components of $G_x$ is abelian, a clear estimate of the domain of convergence is given.

Moreover when $S$ is a hypersurface and the Hasse principle holds for $G$, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that $(G,\rho,X)$ is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

A polynomial $P(w)$ is called a uniqueness polynomial (or a uiqueness polynomial in a broad sense) if $P(f) = cP(g)$ (or $P(f) = P(g)$) implies $f = g$ for any nonzero constant $c$ and nonconstant meromorphic functions $f$ and $g$ on $\mathbf C$. We consider a monic polynomial $P(w)$ without multiple zero whose derivative has mutually distinct $k$ zeros $e_j$ with multiplicities $q_j$. Under the assumption that $P(e_\ell) \not= P(e_m)$ for all distinct $\ell$ and $m$, we prove that $P(w)$ is a uniqueness polynomial in a broad sense if and only if $\sum_{\ell>m}q_\ell q_m > \sum_{\ell} q_\ell$. We also give some sufficient conditions for uniqueness polynomials.

Let $G$ be a connected reductive algebraic group over an algebraic closure of a finite field of characteristic $p$. Under the assumption that $p$ is good for $G$, we prove that for each character sheaf $A$ on $G$ which has non-zero restriction to the unipotent variety of $G$, there exists a unipotent class $C_A$ canonically attached to $A$, such that $A$ has non-zero restriction on $C_A$, and any unipotent class $C$ in $G$ on which $A$ has non-zero restriction has dimension strictly smaller than that of $C_A$.

The main purpose of this paper is to make a systematic study of a special type of conformally Kähler manifolds, called multiplier Hermitian manifolds, which we often encounter in the study of Hamiltonian holomorphic group actions on Kähler manifolds. In particular, we obtain a multiplier Hermitian analogue of Myers' Theorem on diameter bounds with an application (see [M5]) to he uniquness up to biholomorphisms of the "Kähler-Einstein metrics" in the sense of [M1] on a given Fano manifold with nonvanishing Futaki character.

In this paper the authors prove that a class of multilinear operators formed by the singular integral or fractional integral operators with homogeneous kernels are bounded operators from the product spaces $L^{p_1} \times L^{p_2} \times \cdots \times L^{p_K}({\mathbb R}^n)$ to the Hardy spaces $H^q({\mathbb R}^n)$ and the weak Hardy space $H^{q,\infty}({\mathbb R}^n)$, where the kernel functions $\Omega_{ij}$ satisfy only the $L^s$-Dini conditions. As an application of this result, we obtain the $(L^p, L^q)$ boundedness for a class of commutator of the fractional integral with homogeneous kernels and BMO function.

We investigate the $ \Omega_{\pm} $-result of $\Delta_{a}(x)$ and its number of sign-changes in an interval $[1,T]$, denoted by $X_{a}(T)$, for $-1 \le a < -1/2$. We can prove that $T \ll_{a} X_{a}(T)$ which is the best possible in order of magnitude.

For a domain $A$ containing a field $k$ with ${\rm tr.deg}_{k} A < \infty$, we define a new transcendence degree of $A$ with respect to $k$, which is denoted by ${\rm td}_{k} A$. By using this, we generalize the theorem that for every affine domain $A$ over a field $k$ it holds that $\dim A = {\rm tr.deg}_{k} A$. For example, we show that if $A$ is a quasi-local domain containing a field $k$ with $\dim A = {\rm td}_{k} A < \infty$, then for every Noetherian local $k$-subalgebra $R$ of $A$ it holds that $\dim R = {\rm td}_{k} R$. Moreover we also generalize the theorem due to Gilmer, Nashier and Nichols.

In this paper, we deal with the relation between the characteristic functions of meromorphic functions that share three values CM. As applications of our main results, we shall affirmatively settle two conjectures proposed by Mues and Osgood-Yang.

We study the notion of special tight closure of an ideal and show that it can be used as a tool for tight closure computations.

We study the asymptotic distribution of rational points on a generalized flag variety which are of bounded height and satisfy some congruence conditions in the formulation analogous to a strongly Hardy-Littlewood variety.