Related papers: An explicit height bound for the classical modular…

We consider a quasi-homogeneous polynomial $f \in \mathbb{Z}[x_0, \ldots, x_N]$ of degree $w$ equal to the degree of $x_0 \cdots x_N$ and show that the $F$-pure threshold of the reduction $f_p \in \mathbb{F}_p[x_0, \ldots, x_N]$ is equal to…

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and…

Consider the interval of integers $I_{m,n} = \{m, m+1, m+2,\ldots, m+n-1 \}$. For fixed integers $h,k,m$, and $c$, let $\Phi_{h,k,m}^{(c)}(n)$ denote the number of solutions of the equation $(a_1+\cdots + a_h)- (a_{h+1} + \cdots +…

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…

We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ is the height of the polynomial. Next, restricting…

Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…

Let K be a function field in one variable over an arbitrary field F. Given a rational function f(z) in K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic…

We prove that every planar poset $P$ of height $h$ has dimension at most $192h + 96$. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets…

The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that…

For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of…

Let F and G be morphisms of degree at least 2 from P^N to P^N that are defined over the algebraic closure of Q. We define the arithmetic distance d(F,G) between F and G to be the supremum over all algebraic points P of |h_F(P)-h_G(P)|,…

Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these…

For any $m \geq 1$, let $H_m$ denote the quantity $H_m := \liminf_{n \to \infty} (p_{n+m}-p_n)$, where $p_n$ denotes the $n^{\operatorname{th}}$ prime; thus for instance the twin prime conjecture is equivalent to the assertion that $H_1$ is…

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…

Let $f,g \in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\circ h$. The…

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…

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound…

What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3 \cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then $f(n_k) = h_k$…

Define the height function h(a) = min{k+(ka\mod p): k=1,2,...,p-1} for a = 0,1,...,p-1. It is proved that the height has peaks at p, (p+1)/2, and (p+c)/3, that these peaks occur at a= [p/3], (p-3)/2, (p-1)/2, [2p/3], p-3,p-2, and p-1, and…

Let $\mathcal{H}(n)$ be the maximum number of limit cycles that a planar polynomial vector field of degree $n$ can have. In this paper we prove that $\mathcal{H}(n)$ is realizable by structurally stable vector fields with only hyperbolic…