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The problem of this thesis concerns points of small height on affine varieties defined over arbitrary function fields, and is based on published work with Prof. Dragos Ghioca (see [GN20]). The main result is as follows: the points lying outside the largest subvariety defined over the constant field cannot have arbitrarily small height.Prior results of this type include [Ghi09], [Ghi14]. In particular, [Ghi14] answers this question for function fields of transcendence degree 1. It also captures the history of the subject and features an argument that was initially used by the author of this thesis to extend [Ghi14] to varieties defined over function fields of arbitrary (finite) transcendence degree. The content of this thesis and the associated published paper not only extends [Ghi14] to arbitrary transcendence degree, but also provides a sharp lower bound for points which are not contained in the largest subvarietydefined over the constant field. The argument here works directly with the defining polynomials of the variety (compare with [Ghi14]), and the lower bound only depends on their degrees.

We formulate a variant in characteristic p of the Zariski dense orbit conjecture previously posed by Zhang, Medvedev-Scanlon and Amerik-Campana for rational self-maps of varieties defined over fields of characteristic 0. So, in our setting, let K be an algebraically closed field, which has transcendence degree d ≥ 1 over ??. Let X be a variety defined over K, endowed with a dominant rational self-map Φ. We expect that either there exists a variety Y defined over a finite subfield ?? of ?? of dimension at least d + 1 and a dominant rational map τ: X ⤏Y such that τ o ?ᵐ= Fʳ o τ for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field ??, or either there exists α ⋲ X(K) whose orbit under ? is well-defined and Zariski dense in X, or there exists a non-constant ? : X ⤏ ℙ¹ such that ? o ?= ? . We explain why the new condition in our conjecture is necessary due to the presence of the Frobenius endomorphism in case X is isotrivial. Then we prove our conjecture for all regular self-maps on ?ᴺm.

We prove the positive characteristic version of the Dynamical Mordell-Lang Conjecture in two novel cases. Let p be a prime and K a field of characteristic p>0. Let k ∈ ℕ, and let G denote the multiplicative group of K, of dimension k. Let α be an element of G, and V a variety contained in G. Let φ: G →G be a group endomorphism defined over K. We knowφ(x₁,x₂,...,xk)=(x₁^a1,1 x₂^a1,2 ··· xk^a1k , ... , x₁^ak1 x₂^ak2 ··· xk^akk),for some integer exponents aij. In the case where the matrix of exponents, ( aij ) is similar to a single Jordan block, we show that the set S = { n ∈ ℕ double : φ^n(α) ∈ V } is a finite union of arithmetic progressions. When the dimension k = 3, we show S is a finite union of arithmetic progressions for any group endomorphism φ.

Let K be an algebraically closed field, and let C be an irreducible plane curve, defined over the algebraic closure of K(t), which is not defined over K. We show that there exists a positive real number c₀ such that if P is any point on the curve C whose Weil height is bounded above by c₀, then the coordinates of P belong to K.

Let d ≥2 be an integer, let c ∈ ℚ(t) be a rational map, and let f_t(z) = (z^d+t)/z be a family of rational maps indexed by t. For each t = λ algebraic number, we let ĥ_(f_λ)(c(λ)) be the canonical height of c(λ) with respect to the rational map f_λ; also we let ĥ_f(c) be the canonical height of c on the generic fiber of the above family of rational maps. We prove that there exists a constant C depending only on c such that for each algebraic number λ, |ĥ_(f_λ)(c(λ))-ĥ_f(c)h(λ)| ≤C. [Formula missing]This improves a result of Call and Silverman for this family of rational maps.