On the Zeros of Fermat Quotients and Mirimanoff Polynomials

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.

Domain of attraction of normal law and zeros of random polynomials

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

On the average number of sharp crossings of certain Gaussian random polynomials

Lerch’s Theorems over Function Fields

In this work, we state and prove Lerch’s theorems for Fermat and Euler quotients over function fields defined analogously to the number fields. 1. Results The Fermat’s little theorem states that if p is a prime and a is an integer not divisible by p, then ap−1 ≡ 1 mod p. This gives rise to the definition of the Fermat quotient of p with base a, q(a, p) = ap−1 − 1 p , which is an integer. This q...

Bounds of Multiplicative Character Sums with Fermat Quotients of Primes

Given a prime p, the Fermat quotient qp(u) of u with gcd(u, p)= 1 is defined by the conditions qp(u)≡ u p−1 − 1 p mod p, 0≤ qp(u)≤ p − 1. We derive a new bound on multiplicative character sums with Fermat quotients qp(`) at prime arguments `. 2010 Mathematics subject classification: primary 11A07; secondary 11L40, 11N25.