This work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pn, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of algebraic statistics.

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In [2], Dan Anderson and Andrea Frazier introduced a generalized theory of factorization. Given a relation τ on the nonzero, nonunit elements of an integral domain D, they defined a τ-factorization of a to be any proper factorization a = λa1 · · · an where λ is in U (D) and ai is τ-related to aj, denoted ai τ aj, for i not equal to j . From here they developed an abstract theory of factorization that generalized factorization in the usual sense. They were able to develop a number of results analogous to results already known for usual factorization. Our work focuses on the notion of τ-factorization when the relation τ has characteristics similar to those of coprimeness. We seek to characterize such τ-factorizations. For example, let D be an integral domain with nonzero, nonunit elements a, b ∈ D. We say that a and b are comaximal (resp. v-coprime, coprime ) if (a, b) = D (resp., (a, b)v = D, [a, b] = 1). More generally, if ∗ is a star-operation on D, a and b are ∗-coprime if (a, b)∗ = D. We then write a τmax b (resp. a τv b, a τ[ ] b, or a τ∗ b) if a and b are comaximal (resp. v -coprime, coprime, or ∗-coprime).

We consider and solve some open conjectures on the asymptotic behavior of the number of different numbers of the nonassociate factorizations of prescribed minimal length for specific finite factorization domains. The asymptotic behavior will be classified for Cohen-Kaplansky domains in Chapter 1 and for domains of the form R=K+XF[X] for finite fields K and F in Chapter 2. A corollary of the main result in Chapter 3 will determine the asymptotic behavior for Krull domains with finite divisor class group.

In this thesis, we prove unconditional lower bounds on resources needed to compute explicit functions in the following three models of computation: constant-depth boolean circuits, multivariate polynomials over commutative rings and the 'Number on the Forehead' model of multiparty communication. Apart from using tools from diverse areas, we exploit the rich interplay between these models to make progress on questions arising in the study of each of them. / Boolean circuits are natural computing devices and are ubiquitous in the modern electronic age. We study the limitation of this model when the depth of circuits is fixed, independent of the length of the input. The power of such constant-depth circuits using gates computing modular counting functions remains undetermined, despite intensive efforts for nearly twenty years. We make progress on two fronts: let m be a number having r distinct prime factors none of which divides ℓ. We first show that constant depth circuits employing AND/OR/MODm gates cannot compute efficiently the MAJORITY and MODℓ function on n bits if 'few' MODm gates are allowed, i.e. they need size nW&parl0;1s&parl0;log n&parr0;1/&parl0;r-1&parr0;&parr0; if s MODm gates are allowed in the circuit. Second, we analyze circuits that comprise only MOD m gates, We show that in sub-linear size (and arbitrary depth), they cannot compute AND of n bits. Further, we establish that in that size they can only very poorly approximate MODℓ. / Our first result on circuits is derived by introducing a novel notion of computation of boolean functions by polynomials. The study of degree as a resource in polynomial representation of boolean functions is of much independent interest. Our notion, called the weak generalized representation, generalizes all previously studied notions of computation by polynomials over finite commutative rings. We prove that over the ring Zm , polynomials need Wlogn 1/r-1 degree to represent, in our sense, simple functions like MAJORITY and MODℓ. Using ideas from arguments in communication complexity, we simplify and strengthen the breakthrough work of Bourgain showing that functions computed by o(log n)-degree polynomials over Zm do not even correlate well with MODℓ. / Finally, we study the 'Number on the Forehead' model of multiparty communication that was introduced by Chandra, Furst and Lipton [CFL83]. We obtain fresh insight into this model by studying the class CCk of languages that have constant k-party deterministic communication complexity under every possible partition of input bits among parties. This study is motivated by Szegedy's [Sze93] surprising result that languages in CC2 can all be extremely efficiently recognized by very shallow boolean circuits. In contrast, we show that even CC 3 contains languages of arbitrarily large circuit complexity. On the other hand, we show that the advantage of multiple players over two players is significantly curtailed for computing two simple classes of languages: languages that have a neutral letter and those that are symmetric. / Extending the recent breakthrough works of Sherstov [She07, She08b] for two-party communication, we prove strong lower bounds on multiparty communication complexity of functions. First, we obtain a bound of n O(1) on the k-party randomized communication complexity of a function that is computable by constant-depth circuits using AND/OR gates, when k is a constant. The bound holds as long as protocols are required to have better than inverse exponential (i.e. 2-no1 ) advantage over random guessing. This is strong enough to yield lower bounds on the size of an important class of depth-three circuits: circuits having a MAJORITY gate at its output, a middle layer of gates computing arbitrary symmetric functions and a base layer of arbitrary gates of restricted fan-in. / Second, we obtain nO(1) lower bounds on the k-party randomized (bounded error) communication complexity of the Disjointness function. This resolves a major open question in multiparty communication complexity with applications to proof complexity. Our techniques in obtaining the last two bounds, exploit connections between representation by polynomials over teals of a boolean function and communication complexity of a closely related function.

For any ring R (associative with 1) we associate a space X of prime torsion theories endowed with Golan's SBO-topology. A separated presheaf L(-,M) on X is then constructed for any right R-module M$ sb{ rm R}$, and a sufficient condition on M is given such that L(-,M) is actually a sheaf. The sheaf space rm E { buildrel{ rm p} over longrightarrow} X) etermined by L(-,M) represents M in the following sense: M is isomorphic to the module of continuous global sections of p. These results are applied to the right R-module R$ sb{ rm R}$ and it is seen that semiprime rings satisfy the required condition for L(-,R) to be a sheaf. Among semiprime rings two classes are singled out, fully symmetric semiprime and right noetherian semiprime rings; these two kinds of rings have the desirable property of yielding "nice" stalks for the above sheaf.

Thesis (Ph.D.)--University of Delaware, 2010. / Principal faculty advisor: Robert Coulter, Dept. of Mathematical Sciences. Includes bibliographical references.

We use the behavioural approach and commutative algebra to define and characterize poles and zeros of multidimensional (nD) linear systems. In the case of a system with a standard input output structure we provide new definitions and characterizations of system, controllable and uncontrollable zeros and demonstrate strong relationships between the controllable poles and zeros and properties of the system transfer matrix, and we show that the uncontrollable zeros are in fact uncontrollable poles. We also show that we can regard the zero as a form of pole with respect to an additional form of input output structure imposed on the zero output sub-behaviour. In the case when the behaviour has a latent variable description we make a further distinction of the zeros into several other classes including observable, unobservable and invariant zeros. In addition we also introduce their corresponding controllable and uncontrollable zeros, such as the observable controllable, unobservable controllable, invariant controllable, observable uncontrollable, unobservable uncontrollable and invariant uncontrollable etc. We again demonstrate strong relationships between these and other types of zeros and provide physical interpretations in terms of exponential and polynomial exponential trajectories. In the 1D case of a state-space model we show that the definitions and characterizations of the observable controllable and invariant zeros correspond to the transmission zeros and the invariant zeros in the classical 1D framework. This then completes the correspondences between the behavioural definitions of poles and zeros and those classical poles and zeros which have an interpretation in nD.

In this thesis, we introduce the theory of locally nilpotent derivations and use it to compute certain ring invariants. We prove some results about quasi-extensions of derivations and use them to show that certain rings are non-rigid. Our main result states that if k is a field of characteristic zero, C is an affine k-domain and B = C[T,Y] / < T^nY - f(T) >, where n >= 2 and f(T) \in C[T] is such that delta^2(f(0)) != 0 for all nonzero locally nilpotent derivations delta of C, then ML(B) != k. This shows in particular that the ring B is not a polynomial ring over k.

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