Related papers: Double Character Sums with Intervals and Arbitrary…
In this short note, we use Rudnev's point-plane incidence bound to improve some results on conditional expanding bounds for two-variable functions over arbitrary fields due to Hegyv\'ari and Hennecart.
The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary…
We use the theory of resultants of polynomials to study the stability of an arbitrary polynomial over a finite field, that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial…
We use an estimate of Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) on the number of solutions of certain equations involving products and differences of sets in prime finite fields to give an explicit upper bound on trilinear…
A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…
A binary linear error correcting codes represented by two code families Kronecker products sum are considered. The dimension and distance of new code is investigated. Upper and lower bounds of distance are obtained. Some examples are given.…
We estimate the lattice sums arising in the context of the integer point counting in polyhedra.
Let $q$ be a power of a prime and let $\mathbb{F}_q$ be the finite field consisting of $q$ elements. We establish new explicit estimates on Gauss sums of the form $S_n(a) = \sum_{x\in \mathbb{F}_q}\psi_a(x^n)$, where $\psi_a$ is a…
The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide…
We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ ring homomorphisms into an algebraic closure from an \'etale extension of…
In this paper, we study the summability properties of double sequences of real constants which map sequences of random variables to sequences of random variables that are defined on the same probability sample space. We show that a regular…
For a large prime $p$, and a polynomial $f$ over a finite field $F_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of $F_p^*$ containing $H\ge 1$ consecutive values $f(x)$, $x = u+1, \ldots, u+H$,…
We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is valid for all fields. Then we apply the polynomial method to obtain global bounds…
By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. The…
In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the…
We establish new upper bounds about symmetric bilinear complexity in any extension of finite fields. Note that these bounds are not asymptotical but uniform. Moreover we give examples of Shimura curves that do not descend over their field…
Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on…
We give explicit bounds on sums of $d(n)^2$ and $d_4(n)$, where $d(n)$ is the number of divisors of $n$ and $d_4(n)$ is the number of ways of writing $n$ as a product of four numbers. In doing so we make a slight improvement on the upper…
In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new…
In this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer-manin obstruction (see \cite{CTX}). The…