The goal of this thesis is to explore the properties of a certain class of sequences, rigid divisibility sequences, generated by the iteration of certain polynomials whose coefficients are algebraic integers. The main goal is to provide, as far as is possible, a classification and description of those polynomials which generate rigid divisibility sequences.

Let G = (V,A,E) be a mixed graph and co : V → {1, 2,...,λ} a function such that co is a proper colouring of the underlying graph, Und(G), and co(u) ≠ co(y) when co(v) = co(x), for every pair of arcs (u,v) and (x,y). Such a function is called a proper oriented λ − colouring of G. The number of proper oriented λ–colourings of G, denoted fo(G,λ), is a polynomial in λ. We call fo(G,λ) the mixed-chromatic polynomial of G. In this thesis we will first present the basic theory of the mixed-chromatic poly-nomial. This theory will include computational tools and results concerning the coefficients of fo(G,λ). Next, we will consider the question of chromatic uniqueness and invariance of mixed graphs. Lastly, we reformulate a contract-delete recurrence for chromatic polynomials in order to enumerate various colourings, such as k−frugal λ−colourings. / Graduate

Multi-dimensional (n-D) systems can be described by matrices whose elements are polynomial in more than one indeterminate. These systems arise in the study of partial differential equations and delay differential equations for example, and have attracted great interest over recent years. Many of the available results have been developed by generalising the corresponding results from the well known 1-D theory. However, this is not always the best approach since there are many differences between 1-D, 2-D and n-D (n > 2) polynomial matrices. This is due mainly to the underlying polynomial ring structure.

Let p be a prime and q = pk. The polynomial gn,q isin Fp[x] defined by the functional equation Sigmaa isin Fq (x+a)n = gn,q(xq- x) gives rise to many permutation polynomials over finite fields. We are interested in triples (n,e;q) for which gn,q is a permutation polynomial of Fqe. In Chapters 2, 3, and 4 of this dissertation, we present many new families of permutation polynomials in the form of gn,q. The permutation behavior of gn,q is becoming increasingly more interesting and challenging. As we further explore the permutation behavior of gn,q, there is a clear indication that gn,q is a plenteous source of permutation polynomials. We also describe a piecewise construction of permutation polynomials over a finite field Fq which uses a subgroup of Fq*, a “selection” function, and several “case” functions. Chapter 5 of this dissertation is devoted to this piecewise construction which generalizes several recently discovered families of permutation polynomials.

Understanding the structure and complexity of a polynomial family is a fundamental problem of arithmetic circuit complexity. There are various approaches like studying the lower bounds, which deals with nding the smallest circuit required to compute a polynomial, studying the orbit and stabilizer of a polynomial with respect to an invertible transformation etc to do this. We have a rich understanding of some of the well known polynomial families like determinant, permanent, IMM etc. In this thesis we study some of the structural properties of the polyno-mial family called the Nisan-Wigderson polynomial family. This polynomial family is inspired from a well known combinatorial design called Nisan-Wigderson design and is recently used to prove strong lower bounds on some restricted classes of arithmetic circuits ([KSS14],[KLSS14], [KST16]). But unlike determinant, permanent, IMM etc, our understanding of the Nisan-Wigderson polynomial family is inadequate. For example we do not know if this polynomial family is in VP or VNP complete or VNP-intermediate assuming VP 6= VNP, nor do we have an understanding of the complexity of its equivalence test. We hope that the knowledge of some of the inherent properties of Nisan-Wigderson polynomial like group of symmetries and Lie algebra would provide us some insights in this regard. A matrix A 2 GLn(F) is called a symmetry of an n-variate polynomial f if f(A x) = f(x): The set of symmetries of f forms a subgroup of GLn(F), which is also known as group of symmetries of f, denoted Gf . A vector space is attached to Gf to get the complete understanding of the symmetries of f. This vector space is known as the Lie algebra of group of symmetries of f (or Lie algebra of f), represented as gf . Lie algebra of f contributes some elements of Gf , known as continuous symmetries of f. Lie algebra has also been instrumental in designing e cient randomized equivalence tests for some polynomial families like determinant, permanent, IMM etc ([Kay12], [KNST17]). In this work we completely characterize the Lie algebra of the Nisan-Wigderson polynomial family. We show that gNW contains diagonal matrices of a speci c type. The knowledge of gNW not only helps us to completely gure out the continuous symmetries of the Nisan-Wigderson polynomial family, but also gives some crucial insights into the other symmetries of Nisan-Wigderson polynomial (i.e. the discrete symmetries). Thereafter using the Hessian matrix of the Nisan-Wigderson polynomial and the concept of evaluation dimension, we are able to almost completely identify the structure of GNW . In particular we prove that any A 2 GNW is a product of diagonal and permutation matrices of certain kind that we call block-permuted permutation matrix. Finally, we give explicit examples of nontrivial block-permuted permutation matrices using the automorphisms of nite eld that establishes the richness of the discrete symmetries of the Nisan-Wigderson polynomial family.

We use transcendental shift-like automorphisms of Ck, k > 2 to construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of Ck, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon-Esterle in C2. The second example shows the existence of a Fatou-Bieberbach domain in Ck,k > 2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay-Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.

No description available.

Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .

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.