Related papers: Gaps in binary cyclotomic polynomials

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$…

What is the maximum possible value of the lead coefficient of a degree $d$ polynomial $Q(x)$ if $|Q(1)|,|Q(2)|,\ldots,|Q(k)|$ are all less than or equal to one? More generally we write $L_{d,[x_k]}(x)$ for what we prove to be the unique…

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a…

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this…

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that \[ P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}) =Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}). \] We denote this polynomial…

In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also…

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms…

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the…

Let $\mathcal{P}$ be a chiral polytope with type $\{k_1, k_2\}$ and $G=Aut(\mathcal{P})$. Suppose $|G|=2p^m$, where $k_1, k_2\geq 3$ and $p$ is an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes…

The height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic polynomials. One result, due to H. Maier, gives a best…

Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the…

In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a…

In this paper it is proved, that for every prime number p, the set of cyclic p-roots in C^p is finite. Moreover the number of cyclic p-roots counted with multiplicity is equal to (2p-2)!/(p-1)!^2. In particular, the number of complex…

Let F and K be fields of characteristic 0, with F a subset of K. Let K[x] denote the ring of polynomials with coefficients in K. For p in K[x]\F[x], deg(p) = n, let r be the highest power of x with a coefficient not in F. We define the F…

Let $q=p^\alpha$ be a fixed prime power, $k\geq 2$ be an integer. We give a new upper bound for the size of $k$-wise $q$-modular $L$-avoiding $L$-intersecting set systems, where $L$ is any proper subset of $\{0, \ldots , q-1\}$. Our proof…

We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the…

A real-valued sequence $f = \{ f(n) \}_{n \in \mathbb{N}}$ is said to be second-order holonomic if it satisfies a linear recurrence $f (n + 2) = P (n) f (n + 1) + Q (n) f (n)$ for all sufficiently large $n$, where $P, Q \in \mathbb{R}(x)$…

Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We…

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower…