Related papers: Squarefree polynomials with prescribed coefficient…

Let K be a field and let A be the polynomial ring in n variables with coefficients in the field K We study the universal squarefree lexsegment ideals in A. We put our attention on their combinatorics computing some invariants. Moreover we…

We prove that if $f$ is a non zero cusp form of weight $k$ on $\Gamma_0(N)$ with character $\chi$ such that $N/(\text{conductor }\chi)$ square-free, then there exists a square-free $n\ll_{\epsilon} k^{3+\epsilon}N^{7/2+\epsilon}$ such that…

In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever…

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a power of prime $p$. A polynomial over $\mathbb{F}_{q}$ is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of…

Let $g>1$ be an integer and $f(X)\in{\mathbb Z}[X]$ a polynomial of positive degree with no multiple roots, and put $u(n)=f(g^n)$. In this note, we study the sequence of quadratic fields ${\mathbb Q}(\sqrt{u(n)}\,)$ as $n$ varies over the…

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_q$ of $q$ elements. A polynomial in $\mathbb{F}_q[t]$ is called $m$-smooth (or $m$-friable) if all its irreducible factors are of degree at most $m$. In this…

Connection of flat polynomials with some spectral questions in ergodic theory is discussed. A necessary condition for a sequence of polynomials of the type $\frac{1}{\sqrt{N}} \big(1 +\sum_{j=1}^{N-1} z^{n_j}\big)$ to be flat in almost…

We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in \mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as a function…

A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this…

We show that the exponent of distribution of the sequence of squarefree numbers in arithmetic progressions of prime modulus is $\geq 2/3 + 1/57$, improving a result of Prachar from 1958. Our main tool is an upper bound for certain bilinear…

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta <…

Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is square-free.

This is an exposition of results of R.C. Vaughan and the author (Mathematika 70 (2024), no. 4). We discuss how often the squarefree values of an integral polynomial do occur. We discuss interrelations between our results and results of B.…

Let $k\geq 2$ be a square-free integer. We prove that the number of square-free integers $m\in [1,N]$ such that $(k,m)=1$ and $\mathbb{Q}(\sqrt[3]{k^2m})$ is monogenic is $\gg N^{1/3}$ and $\ll N/(\log N)^{1/3-\epsilon}$ for any…

We show that if a polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ is nonnegative on a closed basic semialgebraic set $X=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\ldots,g_r (x)\ge 0\}$, where $g_1,\ldots,g_r\in\mathbb{R}[x_1,\ldots,x_n]$, then $f$ can be…

In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…

We formulate and prove a necessary condition for a sequence of analytic trigonometric polynomials with real non-negative coefficients to be flat a.e.

We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.

Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski…

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…