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In this paper, we prove an extension theorem for spheres of square radii in $mathbb{F}_q^d$, which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem. We also will study applications on distance problems.
Let k be a perfect field of positive characteristic, k(t)_{per} the perfect closure of k(t) and A=k[[X_1,...,X_n]]. We show that for any maximal ideal N of A=k(t)_{per}otimes_k A, the elements in hat{A_N} which are annihilated by the Taylor Hasse-Sch
Let $mathbb{F}_q$ be a finite field of order $q$. Given a set $S$ of oriented spheres in $mathbb{F}_q^d$, how many pairs of spheres can be in contact? In this paper, we provide a sharp result for this question by using discrete Fourier analysis. More
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli sp
We study the lower semicontinuity for functionals defined on compact sets in R^2 with a finite number of connected components and finite length which depend on their normal vector. We apply the result to the study of quasi-static growth of brittle fr
The beautiful Beraha-Kahane-Weiss theorem has found many applications within graph theory, allowing for the determination of the limits of root of graph polynomials in settings as vast as chromatic polynomials, network reliability, and generating pol