Related papers: On the existence of Hurwitz polynomials with no Ha…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ of degree $k$, a family of polynomials $F_m(x)$ depending on $f(x)$ is constructed with the properties: (i) there is exactly one irreducible factor $\Phi_{d,f}(x)$…

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let l>2 be prime and A a finite abelian l-group. Then there…

We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$ where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring…

We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the…

This paper show that two doubly infinite series generate a totally nonnegative Hurwitz-type matrix if and only if their ratio represents an S-functions of a certain kind. The doubly infinite case needs a specific approach, since the ratios…

A polynomial whose coeffcients are equal to its roots is called a Ulam polynomial. In this paper we show that for a given degree n there exists a finite number of Ulam polynomials of degree n.

In this paper we prove certain Hurwitz equivalence properties of $B_n$. In particular we prove that for $n=3$ every two Artin's factorizations of $\Delta _3 ^2$ of the form $H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}$ (with $i_k, j_k…

Sendov's conjecture asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all of its zeroes in closed unit disk $\{ z: |z| \leq 1 \}$, then for each such zero $\lambda_0$ there is a zero of the derivative $f'$ in the closed unit…

We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit…

In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Recently, Branden used this theory and a polynomial coming from the Vamos matroid to disprove the generalized…

We prove that for every smooth Jordan curve $\gamma \subset \mathbb{C}$ and for every set $Q \subset \mathbb{C}$ of six concyclic points, there exists a non-constant quadratic polynomial $p \in \mathbb{C}[z]$ such that $p(Q) \subset…

We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| < 1\}$. For every pair $(k, n) \in…

We investigate the computational problem of determining whether a bivariate polynomial with non-negative coefficients and no constant term can attain a prime value. While classical conjectures such as Bouniakowsky's provide necessary…

If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $\pm1$, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable. In…

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact,…

We prove a Fej\'er-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on $\mathscr{W}\times\mathfrak{Y}$, where $\mathscr{W}$ is either the free semigroup $\langle x \rangle_g$ or the free product…

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

Several recent papers have considered the problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to rings of integers in quadratic number fields…

We prove a conjecture due to Wanless about the permanent of Hadamard matrices in the particular case of Sylvester-Hadamard matrices. Namely we show that for all n greater or equal to 2, the dyadic valuation of the permanent of the…