Tame Decompositions and Collisions

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if $n$ has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all 2-collisions. We introduce a normal form for multi-collisions of decompositions of arbitrary length with exact description of the (non)uniqueness of the parameters. We obtain an efficiently computable formula for the exact number of such collisions at degree n over a finite field of characteristic coprime to p. This leads to an algorithm for the exact number of decomposable polynomials at degree n over a finite field Fq in the tame case

Counting classes of special polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. In two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. We present counting results for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). These numbers come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f over a field F is decomposable if f = g o h with nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of F does not divide n = deg f, is fairly well understood, and the upper and lower bounds on the number of decomposable polynomials of degree n match asymptotically. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. There is an obvious inclusion-exclusion formula for counting. The main issue is then to determine, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs. We provide a normal form for collisions of any number of compositions with any number of components. This generalization yields an exact formula for the number of decomposable polynomials of degree n coprime to p. For the wild case, we classify all collisions at degree n = p^2 and obtain the exact number of decomposable polynomials of degree p^2

Efficient computation of regular differential systems by change of rankings using Kähler differentials

We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated

Efficient computation of regular differential systems by change of rankings using Kähler differentials

We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated

Poles of the resolvent

M.Sc. (Mathematics)Any sensible piece of writing has an intended readership. Conversely, any piece of writing that has no intended readership has no sense. These are axioms of authorship and necessary directions to any prospective author. The aim of this dissertation was to serve as an experimental exposition of the analysis of the resolvent operator. Its intended readership is therefore graduate-level students in operator theory and Banach algebras. The analysis included in this dissertation is of a specific kind: it includes and occasionally extends beyond the analysis of a function at certain of its singularities of finite order. The exposition is experimental in the sense that it does not even aim at a comprehensive review of analysis of the resolvent operator, but it is concerned with that part of it which seems to have interesting and useful results and which appears to be the most suggestive of further research. In order to obtain an exhaustive exposition, we still lack a study of the properties of the resolvent operator where it is differentiable (which seemingly entails little more than undergraduate-level complex analysis), and a study of essential singularities of the resolvent operator (which seems too difficult for the expository style). A brief overview of the contents of this dissertation is in order: a chapter introducing some analytic concepts used throughout this dissertation; a chapter on poles of order 1 follows (so-called simple poles), where the Gelfand theorem (2.1.1) is the most important result; a chapter on poles of higher order, where the Hille theorem is the most prominent; and lastly some topics that have arisen out of the study of poles of the resolvent, collected in chapter 4. I should make it abundantly clear to the reader that although this dissertation is my work, it does not for the most part follow that the result are my own. What is my own is the arrangement, but as it is a literature study, the results are mainly those of other authors. My own addition has been mostly notes, usually in italics. The literature study has benefited very much from Zemanek's paper (Zemanek,[54]), and I am deeply indebted to him for it. Incidentally, this has also been a chance to exhibit my style of citation; the number corresponds to the number of the citation in the bibliography. There are numerous instances where I have indicated possible extensions and recumbent studies that could be roused effectively, but which have swelled this volume unnecessarily. For instance, the last subsection is little more than such indications

The Gamut: A Journal of Ideas and Information, No. 35, Spring 1992

CONTENTS OF ISSUE NO. 35, SPRING, 1992 Editorial Louis T. Milic: IT’s the Thought That Counts, 3 Civil Rights Jonathan L. Entin: Going Around and Coming Around in Prince Edward County, 5 An ironic twist in the long road to desegregation Animal Rights Pamela Harrison: Saving Dolphins, Not Eating Meat, 15 A moral basis for vegetarianism Robert J. White: The Animal Rights Movement: A New Pseudo-Religion Activists sabotage medical research Ted Bartlett: Animals are Not People Whose best interest is involved? China Peter Scheckner: American Movies in China, 30 E.T. and Crocodile Dundee in Tiananmen Square Mathematics Jack Soules: My Road to Goldbach’s Conjecture, 41 The fascinating properties of whole numbers Poetry Barbara Moore: Primary Colors, 50 The Premonition, 51 Leonard Trawick: Editor’s Choice: A Commentary, 52 Indian Art Thomas Eugene Donaldson: Varahi and Chamunda: Two Terrifying Females, 55 Indians goddesses embody human fears Languages of the World Esmeralda Manandise: Basque: the Language of the Angels, 65 The language the Devil couldn’t learn Fiction William Feuer: The Writer’s Model, 74 Religion Steven Schmidt: A New Human Testament, 87 A new world religion may be the best hope for world peacehttps://engagedscholarship.csuohio.edu/gamut_archives/1032/thumbnail.jp