# Browsing by Subject "Polynomials"

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Item A group of families of polynomials with imaginary zeros(Texas Tech University, 1938-08) McClain, Elmer CarlShow more Not availableShow more Item Algebraic and analytic techniques in coding theory(2015-12) Bhowmick, Abhishek; Zuckerman, David I.; Bajaj, Chandrajit; Gal, Anna; Lovett, Shachar; Price, EricShow more Error correcting codes are designed to tackle the problem of reliable trans- mission of data through noisy channels. A major challenge in coding theory is to efficiently recover the original message even when many symbols of the received data have been corrupted. This is called the unique decoding problem of error correcting codes. More precisely, if the user wants to send K bits, the code stretches K bits to N bits to tolerate errors in the N bits. Then the goal is to recover the original K bits of the message. Often, the receiver requires only a certain part of the message. In such cases, analyzing the entire received data (word) becomes prohibitive. The challenge is to design a local decoder which queries only few locations of the received word and outputs the part of the message required. This is known as local decoding of an error correcting code. The unique decoding problem faces a certain combinatorial barrier. That is, there is a limit to the number of errors it can tolerate in order to uniquely identify the correct message. This is called the unique decoding radius. A major open problem is to understand what happens if one allows for errors beyond this threshold. The goal is to design an algorithm that can recover the right message, or possibly a list of messages (preferably a small number). This is referred to as list decoding of an error correcting code. At the core of many such codes lies polynomials. Polynomials play a fundamental role in computer science with important applications in algorithm design, complexity theory, pseudo-randomness and machine learning. In this dissertation, we improve our understanding of well known classes of codes and discover various properties of polynomials. As an additional consequence, we obtain results in a suite of problems in effective algebraic geometry, including Hilbert’s nullstellensatz, ideal membership problem and counting rational points in a variety.Show more Item Algebraic points of small height with additional arithmetic conditions(2004) Fukshansky, Leonid Eugene; Vaaler, Jeffrey D.Show more Item Auxiliary polynomials and height functions(2007) Samuels, Charles Lloyd, 1980-; Vaaler, Jeffrey D.Show more We establish two new results in this dissertation. Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number [alpha] under certain assumptions on [alpha]. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. In particular, we prove the following theorem. [Mathematical equations]Show more Item Computing the Tutte Polynomial of hyperplane arrangements(2009-08) Geldon, Todd Wolman; Villegas, Fernando RodriguezShow more We are studying the Tutte Polynomial of hyperplane arrangements. We discuss some previous work done to compute these polynomials. Then we explain our method to calculate the Tutte Polynomial of some arrangements more efficiently. We next discuss the details of the program used to do the calculation. We use this program and present the actual Tutte Polynomials calculated for the arrangements E6, E7, and E8.Show more Item Constructing polynomials over finite fields(Texas Tech University, 1988-12) Stamp, Mark StevenShow more In this paper we present an algorithm for constructing a polynomial system corresponding to a given directed labeled graph. Three specific classes of polynomial systems are considered in detail. Then we briefly consider some desirable properties for output sequences and we show how to construct a polynomial system that will produce a given output sequence of zeros and ones of length 2^n.Show more Item Controlling zeros of interpolating series(Texas Tech University, 2001-05) Robinson, Jeffrey N.Show more Some interesting problems arise when classical complex analysis techniques are applied to digital filter theory. Polynomials used in the interpolation of digital signals are called interpolating polynomials. These pol5momials may require modification to assure the convergence of their reciprocals on the unit circle. Such modifications were a principal concern of an earlier paper by R. Barnard, W. Ford and Y. Wang [4]. The distribution of zeros and the orthogonality property of {sinc(m>2:)} enables the construction of an infinite interpolating series for digital signals to which classical results can be applied [4]. For practical purposes it is convenient to consider finite truncations of the infinite series PN- A property that was observed in [4] was that the polynomials obtained by truncating the interpolating series P^ had the property that all of their zeros lie on the unit circle. Natural generalizations of the sine functions are the Bessel functions, Gegenbauer and Jacobi polynomials, and polynomials generated by certain measures. In this paper, we consider polynomials which are obtained by truncating infinite series which are generalizations of the interpolating series P^. We show that these polynomials do not have the property that their zeros lie on the unit circle when the infinite series is based on one of the above natural generalizations of the sine functions.Show more Item The distribution of roots of certain polynomial(2010-05) Rodríguez, Miguel Antonio, 1972-; Villegas, Fernando Rodriguez; Duke, William; Vaaler, Jeff; Voloch, Felipe; Keel, SeanShow more Abstract not available.Show more Item The height in terms of the normalizer of a stabilizer(2008-05) Garza, John Matthew, 1975-; Vaaler, Jeffrey D.Show more This dissertation is about the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We investigate connections between the normalizer of a stabilizer and lower bounds for the Weil height of algebraic numbers. In the Archimedean case we extend a result of Schinzel [Sch73] and in the non-archimedean case we establish a result related to work of Amoroso and Dvornicich [Am00a]. We establish that amongst all polynomials in Z[x] whose splitting fields are contained in dihedral Galois extensions of the rationals, x³-x-1, attains the lowest Mahler measure different from 1.Show more Item Least-squares polynomial curve-fitting utilizing orthogonal polynomials(Texas Tech University, 1966-05) Knight, Robert EdwardShow more Not availableShow more Item Location of zeros of polynomials(Texas Tech University, 1931-08) Cross, James HollieShow more Not availableShow more Item Mahler measure evaluations in terms of polylogarithms(2004) Condon, John Donald; Rodríguez Villegas, FernandoShow more We prove a conjecture of Boyd by showing that the logarithmic Mahler measure of a certain integer polynomial in three variables is equal to 28 ζ(3). 5π2 The proof proceeds by expressing the Mahler measure as a combination of integrals of one-variable logarithmic forms, evaluating these in terms of poly- logarithm functions at algebraic arguments, and using identities to simplify the expression. Next, we indicate how the techniques used in the previous example can be applied to give Mahler measure evaluations in terms of polylogarithms for two families of three variable polynomials. As an example, we work out the details for a four-parameter subfamily. Finally, we discuss an alternative, more algebraic approach to this sort of calculation. This method, developed by Rodriguez Villegas, Boyd, Maillot and others, relies on showing that certain elements of algebraic K-groups are equal to zero. We reinterpret our original problem in this context and consider attempts at its resolution.Show more Item Monotonio polynomial approximation(Texas Tech University, 1971-05) Whitmore, Roy WalterShow more Not availableShow more Item Multiplicative distance functions(2005) Sinclair, Christopher Dean; Vaaler, Jeffrey D.Show more We generalize Mahler’s measure to create the class of multiplicative distance functions on C[x]. These functions are uniquely determined by their action on the roots of polynomials. We find a simple asymptotic condition that determines which functions on C are induced by multiplicative distance functions, and use this to give several examples. In particular, we show how Mahler’s measure restricted to the set of reciprocal polynomials may be viewed as a multiplicative distance function: the reciprocal Mahler’s measure. We then turn to potential theory to demonstrate how new multiplicative distance functions may be created by generalizing Jensen’s formula. In so doing we will introduce multiplicative distance functions which measure the complexity of polynomials in C[x] by comparing the geometry of their roots to compact subsets of C. Let s be a complex variable, and let N be a positive integer. To every multiplicative distance function Φ we will define an analytic function FN (Φ; s) (HN (Φ; s) resp.) which encodes information about the range of values Φ takes on degree N polynomials in R[x] (C[x] resp.). These functions are analytic in the half plane (s) > N. We show that HN (s) can be represented as the determinant of a Gram matrix in a Hilbert space dependent on s and Φ. This revelation allows us to write HN (s) as the product of the norms of N vectors in the associated Hilbert space. Several examples are presented. Similarly, when N is even we introduce a skew-symmetric inner product associated to Φ and s and show that FN (s) can be written as the Pfaffian of an antisymmetric Gram matrix defined from this skew-symmetric inner product. This allows us to write FN (s) as a product of N/2 simpler functions of s. We use this information to compute FN (s) for the reciprocal Mahler’s measure, and in so doing discover that this function is an even rational function of s with rational coefficients and simple poles at small integers.Show more Item On algebraic aspects of control(Texas Tech University, 1983-12) Bailey, Susan GruenhagenShow more Not availableShow more Item On Galois theory and the insolvability of the Quintic(Texas Tech University, 1998-05) Barnes, Marla ChristineShow more Not availableShow more Item On properties of the zeros of the Cesàro approximants to outer functions(Texas Tech University, 1998-05) Wheeler, WilliamShow more In this paper, we will be concerned with determining properties of the zeros of the Cesaro sums for a fairly general class of outer functions. The class of outer functions that we will consider have positive monotonically decreasing coefficients. Let M be the set of functions analytic in D with positive monotonically decreasing coefficients in their series expansion.Show more Item On the suitability of power functions as S-boxes for symmetric cryptosystems(2006-05) Jedlicka, David Charles, 1978-; Voloch, José FelipeShow more I present some results towards a classification of power functions that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over F2n for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [20] on absolutely irreducible curves shows that a monomial x m is not APN over F2n for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over F2. I will show that the latter polynomial’s singularities imply that except in five cases, all power functions have such a factor. Three of these cases are already known to be APN for infinitely many fields. The last two cases are still unproven. Some specific cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.Show more Item On zeros of partial sums of a mapping of the unit disk onto a sector(Texas Tech University, 1989-08) Wang, Hsing YongShow more This paper gives a mapping of the unit disk onto a sector. There exist neighborhoods of the origin such that power series expansions of the mapping in terms of classical Gegenbauer polynomials, Mittag-Leffler polynomials, and hypergeometric functions were obtained. Each such expansion represents the mapping in a disk of appropriate radius. Also, the zero-free region of the partial sums of the mapping when the sectorial angle is maximum was studied.Show more Item Polynomials that arise in a Polya urn gambling game(Texas Tech University, 2005-05) Hanlon, Bret M.Show more A gambling game based on the Polya urn process is discussed. Working with the expected value of the game produces an interesting sum. The main result of the paper is this sum is a polynomial with degree equal to the initial number of balls in the urn.Show more