Related papers: Heights of Function Field Points on Curves Given b…

Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}.…

We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $\epsilon>0$ such that only finitely many $P\in K^{\text{ab}}$ have…

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a…

In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for…

We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those…

We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebicheff polynomials having coefficients in the…

Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a…

If g and h are functions over some field, we can consider their composition f = g(h). The inverse problem is decomposition: given f, determine the ex- istence of such functions g and h. In this thesis we consider functional decom- positions…

We give a criterion when a polynomial $x^n-g$ is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic.

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert--Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a…

Let $X$ be a smooth projective variety over $ \overline{\mathbb Q}$, and $f:X -rightarrow X$ be a dominant rational map. Let $\delta_{f}$ be the first dynamical degree of $f$ and $h_{X}:X( \overline{\mathbb Q})\to [1,\infty)$ be a Weil…

Consider a modular curve $X_G$ defined over a number field $K$, where $G$ is a subgroup of $GL_2(\mathbb{Z}/N\mathbb{Z})$ with $N>2$. The curve $X_G$ comes with a morphism $j: X_G\to \mathbb{P}^1_K=\mathbb{A}^1_K \cup\{\infty\}$ to the…

Fix a hyperelliptic curve $C/\mathbb{Q}$ of genus $g$, and consider the number fields $K/\mathbb{Q}$ generated by the algebraic points of $C$. In this paper, we study the number of such extensions with fixed degree $n$ and discriminant…

In the present article we describe a class of algebraic curves on which rational functions of two arguments may reach all their possible limiting values. We also solve a similar question for functions that can be represented as a uniform…

We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field F_q, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of…

Let K be a field of positive characteristic. When V is a linear variety in K^n and G is a finitely generated subgroup of K^*, we show how to compute the intersection of V and G^n effectively using heights. We calculate all the estimates…

Let $k$ be a perfect field of characteristic $\neq 2$. We prove that the Schmidt rank (also known as strength) of a quartic polynomial $f$ over $k$ is bounded above in terms of only the Schmidt rank of $f$ over $\overline{k}$, an algebraic…

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f=g(h) over finite fields…

Consider a polynomial vector field $\xi$ in $\mathbb{C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of…