Related papers: Generic points on exponential curves
In this paper we introduce a version of irreducible Laguerre polynomials in two variables and prove for it a congruence property, which is similar to the one obtained by Carlitz for the classical Laguerre polynomials in one variable.
For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in…
We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots,…
Two polynomials $F_k(X_1,\dots,X_k)$ and $\Theta_k(X_1,\dots,X_k)$ over $\Bbb F_2$ arose from the study of a conjecture by C. Carlet about the sum-freedom of the multiplicative inverse function of $\Bbb F_{2^n}$. Both $F_k$ and $\Theta_k$…
We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface…
The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;C) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N…
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…
We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms…
The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all…
Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate Calabi-Yau threefolds. A conjectural formula for E-polynomials is derived from the Gromov-Witten theory of local…
This paper investigates whether or not polynomials that are irreducible over $\mathbb{Q}$ and $\mathbb{Z}$ can remain irreducible under substitution by all quadratic polynomials. It answers this question in the negative in the degree 2 and…
We provide two new bounds on the number of visible points on exponential curves modulo a prime for all choices of primes. We also provide one new bound on the number of visible points on exponential curves modulo a prime for almost all…
In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid…
In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…
Combining a modular approach to the $abc$ conjecture developed by the second author with the classical method of linear forms in logarithms, we obtain improved unconditional bounds for two classical problems. First, for Szpiro's conjecture…
We show that every finite type polarised curve in the conformal $2$-sphere with a polynomial conserved quantity admits a resonance point, under a non-orthogonality assumption on the conserved quantity. Using this fact, we deduce that every…
C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Assuming just a few elementary facts in field theory and the exclusion-inclusion formula, we show how one see the…
In the moduli space of complex cubic polynomials with a marked critical point, given any p>=1, we prove that the loci formed by polynomials with the marked critical point periodic of period p is an irreducible curve. Thus answering a…
Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $\mu$ is a zero of $X_n$ if and only if the triple $(\mu,\mu+m,\mu+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the…
Let $F_q$ be a finite field of characteristic $p=2,3$. We give the number of irreducible polynomials $x^m+a_{m-1}x^{m-1}+...+a_0\in\F_q[x]$ with $a_{m-1}$ and $a_{m-3}$ prescribed for any given $m$ if $p=2$, and with $a_{m-1}$ and $a_1$…