Related papers: Short sums of trace functions over function fields…

A function from $\Bbb F_{2^n}$ to $\Bbb F_{2^n}$ is said to be {\em $k$th order sum-free} if the sum of its values over each $k$-dimensional $\Bbb F_2$-affine subspace of $\Bbb F_{2^n}$ is nonzero. This notion was recently introduced by C.…

We formulate a form of square-root cancellation for the operator which sums a mean-zero function over a hyperplane in $R^d$ for $R$ a possibly noncommutative finite ring. Using an argument of Hart, Iosevich, Koh, and Rudnev, we show that…

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect…

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice…

We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion…

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is to bound the maximum of the absolute value of the incomplete sum \[ M(t):=\max_{0\leq…

We show that it is possible for a square integrable function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space. A consequence in the…

We study irreducible restrictions from modules over alternating groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. This is known when the…

We consider classes $ \mathcal{A}_M(S) $ of functions holomorphic in an open plane sector $ S $ and belonging to a strongly non-quasianalytic class on the closure of $ S $. In $ \mathcal{A}_M(S) $, we construct functions which are flat at…

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As…

Our main result establishes functorial desingularization of noetherian quasi-excellent schemes over $\bfQ$ with ordered boundaries. A functorial embedded desingularization of quasi-excellent schemes of characteristic zero is deduced.…

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an n-th root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a…

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational…

I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the…

For a scalar CFT with a monodromy defect, a `subtracted Green function' is derived in terms of an Appell $F_1$ function. A conjectured relation of Gimenez-Grau and Liendo is thereby proved and extended and shown to hold for generalised free…

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

Let $\succsim$ be a binary relation on the set of simple lotteries over a countable outcome set $Z$. We provide necessary and sufficient conditions on $\succsim$ to guarantee the existence of a set $U$ of von Neumann--Morgenstern utility…

We give a new proof of the Hansen-Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This…

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial…

In the theory of approximation there are some problems on approximation of compacts in functional spaces by nonlinear families : first we deal with the polynomial case, and then we consider the analytic case. We demonstrate a negative…