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We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are…

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg\H{o}'s theorem. As a by-product, we also obtain an elementary proof of the…

We show that a period polynomial introduced by the Lehmers coincides with a generalized Wilf polynomial.

We observe that the linearization coefficients for ultraspherical polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra…

Two purposes will be shown in this paper. The first one is to extend the classic Tumura-Clunie type theorem for meromorphic functions of one complex variable to meromorphic functions of several complex variables by using Clunie lemma. The…

We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of \emph{uniform dual ergodicity} for a very large class of dynamical systems with…

Let $q$ be a prime power. We construct stable polynomials of the form $b^{m-1}(x+a)^m+c(x+a)+d$ over a finite field $\mathbb{F}_{q}$ for $m=2,3,4$ by Capelli's lemma. When $m=3$ and $q$ is even, we confirm the conjecture of Ahmadi and…

In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid…

We prove that the set of large values of the trigonometric polynomial over a subset of density of the primes has some additive structure, similarly to what happens for subsets of densities in $\mathbb{Z}/{N}\mathbb{Z}$ but in a weaker form.…

In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…

We find analogues of the primitive divisor results of Zsigmondy, Bang, Bilu-Hanrot-Voutier, and Carmichael in polynomial rings, following the methods of Carmichael.

We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.

We establish two direct extensions to the Butterfly Theorem on the cyclic quadrilateral along with the proofs using the projective method and analytic geometry of the Cartesian coordinate system.

We obtain new partial results supporting the spectral set conjecture in dimension 1.

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

In this note, we use the concept of a polynomial ring to give an elementary proof to Cayley-Hamilton Theorem. We also give an elementary proof to Birkhoff theorem on Bi-stochastic matrices.

We complete the study of some periods of polynomials in (n+1) variables with (n+2) monomials in computing the behavior of these periods in the natural parameter for such a polynomial.