Related papers: On Hall's conjecture

We show that every positive integer different from $3$ and $5$ can be realized as the $m$-invariant of a field.

The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections…

We prove the following statement. Let $f\in\mathbb{R}[x_1,\ldots,x_d]$, for some $d\ge 3$, and assume that $f$ depends non-trivially in each of $x_1,\ldots,x_d$. Then one of the following holds. (i) For every finite sets…

In this note, we prove that if $A$ is a finite set of real numbers such that $|AA| = K|A|$, then for every polynomial $f \in \mathbb{R}[x,y]$ we have that $|f(A,A)| = \Omega_{K,\operatorname{deg} f}(|A|^2)$, unless $f$ is of the form…

We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.

Let $f$ be a polynomial of degree $d>6$, with integer coefficients. Then the paucity of non-trivial positive integer solutions to the equation $f(a)+f(b)=f(c)+f(d)$ is established. The corresponding situation for equal sums of three like…

This short paper is concerned with polynomial Pell equations \[P^2-DQ^2=1,\] with $P,Q,D\in\Bbb C[X]$ and ${deg}(D)=2$. The main result shows that the polynomials $P$ and $Q$ are closely related to Chebyshev polynomials. We then investigate…

The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We…

We prove that for every positive integer $d \ge 2$ there exist polynomial functions $F_d, G_d: \mathbb{N} \to \mathbb{N}$ such that for each positive integer $r$, every order-$d$ tensor $T$ over an arbitrary field and with partition rank at…

We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\tilde{c}$ positive and $\tilde{p}$ negative roots counted with multiplicity, where…

Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\…

Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain…

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of…

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$…

We consider univariate real polynomials with all roots real and with two sign changes in the sequence of their coefficients which are all non-vanishing. One of the changes is between the linear and the constant term. By Descartes' rule of…

I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, i prove that t is greater or equal than 2c+1, if d is odd and t is greater or equal…

Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…

Let $a,b$ and $n$ be positive integers with $a>b$. In this note, we prove that $$(2bn+1)(2bn+3){2bn \choose bn}\bigg|3(a-b)(3a-b){2an \choose an}{an\choose bn}.$$ This confirms a recent conjecture of Amdeberhan and Moll.

We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.