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Given an o-minimal structure on the real field, we consider an elementary extension to a non-archimedean field R, and interpret the algebraically closed field K=R[sqrt(-1)] on this extension. We construct two pregeometries on K: one by considering images under C-definable holomorphic functions, and the other by considering images under proper restrictions of C-definable holomorphic functions together with algebraic functions (i.e. zeros of polynomials).We show that these two pregeometries are the same, generalising a result of A. Wilkie for complex holomorphic functions. We also do some work towards generalising another result of his on local definability of complex holomorphic functions to our non-archimedean setting.

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The geometric properties of the complex numbers and the quaternions, particularly their behavior under fractional linear transformations, make them very useful for modeling certain types of geometric objects. In this thesis, we will connect the characteristics of fractional linear transformations of both the complex numbers and quaternions to the problem of developing and modifying discrete surfaces for problems in computer graphics and engineering.

In this thesis, we put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. Our results generalize a number of previously known theorems, as well as implying many new corollaries with hypotheses concerning monotonicity of the modulus, real, as well as real and imaginary parts of the coefficients separately. We worked with Enestr\"{o}m-Kakeya type hypotheses, yet we were only concerned with the number of zeros of the polynomial. We considered putting the same type of restrictions on the coefficients of three different types of polynomials: polynomials with a monotonicity``flip" at some index $k$, polynomials split into a monotonicity condition on the even and odd coefficients independently, and ${\cal P}_{n,\mu}$ polynomials that have a gap in between the leading coefficient and the proceeding coefficient, namely the $\mu^{\mbox{th}}$ coefficient.

In this thesis, we explore the effect of restricting the coefficients of polynomials on the bounds for the number of zeros in a given region. The results presented herein build on a body of work, culminating in the generalization of bounds among three classes of polynomials. The hypotheses of monotonicity on each class of polynomials were further subdivided into sections concerning r reversals among the moduli, real parts, and both real and imaginary parts of the coefficients.

The classical Lu Qi-keng Conjecture asks whether the Bergman kernel function for every domain is zero free. The answer is no, and several counterexamples exist in the literature. However, the more general Lu Qi-keng Problem, that of determining which domains in Cn have vanishing kernels, remains a difficult open problem in several complex variables. A challenge in studying the Lu Qi-keng Problem is that concrete formulas for kernels are generally difficult or impossible to compute. Our primary focus is on developing methods of computing concrete formulas in order to study the Lu Qi-keng Problem. The kernel for the annulus was historically the first counterexample to the Lu Qi-keng Conjecture. We locate the zeros of the kernel for the annulus more precisely than previous authors. We develop a theory giving a formula for the weighted kernel on a general planar domain with weight the modulus squared of a meromorphic function. A consequence of this theory is a technique for computing explicit, closed-form formulas for such kernels where the weight is associated to a meromorphic kernel with a finite number of zeros on the domain. For kernels associated to meromorphic functions with an arbitrary number of zeros on the domain, we obtain a weighted version of the classical Ramadanov's Theorem which says that for a sequence of nested bounded domains exhausting a limiting domain, the sequence of associated kernels converges to the kernel associated to the limiting domain. The relationship between the zeros of the weighted kernels and the zeros of the corresponding unweighted kernels is investigated, and since these weighted kernels are related to unweighted kernels in C^2, this investigation contributes to the study of the Lu Qi-keng Problem. This theory provides a much easier technique for computing certain weighted kernels than classical techniques and provides a unifying explanation of many previously known kernel formulas. We also present and explore a generalization of the Lu Qi-keng Problem.

In this thesis we study complex harmonic functions of the form f where f is the sum of a nonconstant analytic and a nonconstant anti-analytic function of one variable. The Fundamental Theorem of Algebra does not apply to such functions, so we ask how many zeros a complex harmonic function can have and where those zeros are located. This thesis focuses on the complex harmonic family of polynomials p_c where p_c is the sum of z+(c/2)z^2 and the conjugate of (c/(n-1))z^(n-1)+(1/n)z^n. We first establish properties of the critical curve, which separates orientation preserving and reversing regions. These properties are then used to show the sum of the orders of the zeros of p_c is -n. In turn, we use this to show p_c has n+2 zeros when 04 and n+4 zeros when c>4, n>5. The total number of zeros of p_c changes when zeros interact with the critical curve, so we investigate where zeros occur on the critical curve to understand how the number of zeros of p_c changes for c between 1 and 4.

Doctor of Philosophy / Department of Mathematics / Pietro Poggi-Corradini / We consider iteration of an analytic self-map f of the unit ball in the N-dimensional complex space C[superscript]N. Many facts were established about such maps and their dynamics in the 1-dimensional case (i.e. for self-maps of the unit disk), and we generalize some of them in higher dimensions. In one dimension, the classical Denjoy-Wolff theorem states the convergence of forward iterates to a unique attracting fixed point, while backward iterates have much more complicated nature. However, under additional conditions (when the hyperbolic distance between two consecutive points stays bounded), backward iteration sequence converges to a point on the boundary of the unit disk, which happens to be a fixed point with multiplier greater than or equal to 1. In this paper, we explore backward-iteration sequences in higher dimension. Our main result shows that in the case when f is hyperbolic or elliptic, such sequences with bounded hyperbolic step converge to a point on the boundary, other than the Denjoy-Wolff (attracting) point. These points are called boundary repelling fixed points (BRFPs) and possess several nice properties. In particular, in the case when such points are isolated from other BRFPs, they are completely characterized as limits of backward iteration sequences. Similarly to classical results, it is also possible to construct a (semi) conjugation to an automorphism of the unit ball. However, unlike in the 1-dimensional case, not all BRFPs are isolated, and we present several counterexamples to show that. We conclude with some results in the parabolic case.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.