Related papers: Squareful numbers in hyperplanes

A square-free integer is a positive integer that is not divisible by the square of any prime. Merten's function, $M(x)$ is defined as the difference between the number of square free integers with an even number of prime factors and the…

In [Sharir and Solomon 2015], Sharir and Solomon showed that the number of incidences between $m$ distinct points and $n$ distinct lines in $\mathbb R^4$ is $$O^*\left(m^{2/5}n^{4/5}+ m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + m +…

Let $\epsilon > 0$ be sufficiently small and let $0 < \eta < 1/522$. We show that if $X$ is large enough in terms of $\epsilon$ then for any squarefree integer $q \leq X^{196/261-\epsilon}$ that is $X^{\eta}$-smooth one can obtain an…

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer…

A nonnegative number d_infinity, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and…

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that…

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result…

We establish the asymptotic formula for the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups over global function fields, given by the sum of the products of local…

We derive asymptotic formulas with a secondary term for the (smoothly weighted) count of number of integer solutions of height $\leqslant B$ with local conditions to the equation $F(x_1,x_2,x_3)=m$, where $F$ is a non-degenerate indefinite…

For nonempty subsets $S_0, \dots, S_{n-1}$ of a (large enough) finite field $\mathbb{F}$ satisfying $$|S_1|, \dots, |S_{n-1}| > 2 \quad \mathrm{or} \quad |S_1|,|S_{n-1}| > n - 1,$$ we show that there exist $a_0 \in S_0, \dots, a_{n-1} \in…

This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of…

We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such…

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For…

The Cayley cubic surface is given by the equation sum_{i=1}^4 X_i^{-1}=0. We show that the number of non-trivial primitive integer points of size at most B is of exact order B(log B)^6, as predicted by Manin's conjecture.

We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free…

For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a…

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k. \] When $n\geq4$ the previous best result is by Baragar (1998) that…

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out…

Let $f$ be an algebraically nondegenerate meromorphic mapping from $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and let $Q_1,...,Q_q$ be $q$ hypersurfaces in $\mathbb P^n(\mathbb C)$ of degree $d_i$, in $N-$subgeneral position. In this…

Using the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and…