Related papers: Squareful numbers in hyperplanes

There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more…

Let M be a singular irreducible complex manifold of dimension n. There are Q divisors D[-1], D[0], D[1],...,D[n+1] on Nash's manifold U -> M such that D[n+1] is relatively ample on bounded sets, D[n] is relatively eventually basepoint free…

Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two…

Let $r \ge 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$, up to $x$, for which the products $\prod_{p \mid n} (p+a)$ (over primes $p$) are…

Let $X$ be a connected compact 3-manifold with non-empty boundary. Consider the boundary $M$ of $X\times D^2$. $M$ is a 4-dimensional closed manifold and has the same fundamental group as $X$. Various examples of $X$ are known for which a…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…

A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of…

Erd\H{o}s asked what is the maximum number $\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point…

In this paper, we construct a family of asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to -6. The manifolds we constructed can be arbitrary close to anti-de Sitter-Schwarzschild manifolds at infinity.…

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $\omega_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems…

Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x n matrices over {0,1,2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x…

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$.…

We consider maximal kernel-operators on abstract measure spaces $(X,\mu)$ equipped with a ball-basis. We prove that under certain asymptotic condition on the kernels those operators maps boundedly BMO(X) into BLO(X), generalizing the…

The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real…

We show that, unlike del Pezzo surfaces, higher dimensional Fano manifolds do not satisfy in general boundedness properties for their ${\rm CH}_0$ group of $0$-cycles. For example, for quartic threefolds having a point of odd degree, there…

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…

Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various…

Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive…

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed…

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets…