Related papers: Counting integer points on affine surfaces with a …
In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These…
We provide an estimate for the number of nontrivial integer points on the Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on the size of fundamental solutions for almost all $d$ in a certain class, based on a…
We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
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…
Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of…
The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the…
We prove effective Pila--Wilkie estimates for the number of rational points of bounded height lying on certain surfaces defined by Pfaffian functions. The class of surfaces to which our result applies includes, for instance, graphs of…
We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
We suggest an upper bound on binomial coefficients that holds over the entire parameter range and whose form repeats the form of the de Moivre-Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the…
We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$.…
Illumination of scenes is usually generated in computer graphics using polygonal meshes. In this paper, we present a geometric method using projections. Starting from an implicit polynomial equation of a surface in 3-D or a curve in 2-D, we…
We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those…
In this article we prove several new uniform upper bounds on the number of points of bounded height on varieties over $\mathbb{F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on $q$ and the…
We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a…
In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a…
Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by…
We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this…