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On Computing with Perron Numbers: a Summary of New Discoveries in Cyclotomic Perron Numbers and New Computer Algo- Rithms for Continued Research

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On Computing with Perron Numbers: a Summary of New Discoveries in Cyclotomic Perron Numbers and New Computer Algo- Rithms for Continued Research On Computing with Perron Numbers: A Summary of New Discoveries in Cyclotomic Perron Numbers and New Computer Algo- rithms for Continued Research A Thesis Presented to the Honors Tutorial College at Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with a Bachelor of Science Degree in Mathematics (BS 1903) and a Bachelor of Science Degree in Computer Science (BS 1929) By William C. Kanieski May 3, 2021 c 2021 William C. Kanieski. All Rights Reserved. This thesis has been approved by the Ohio University Mathematics Department, the School of Electrical Engineering and Computer Science, and the Honors Tutorial College. Dr. Alexei Davydov Director of Studies, Mathematics Thesis Adviser Dr. David Chelberg Director of Studies, Computer Science Thesis Adviser Dr. Donal Skinner Dean, Honors Tutorial College 1 On Computing with Perron Numbers William Kanieski May 3, 2021 Contents I The Mathematics of Cyclotomic Perron Numbers 4 1 Introduction to Cyclotomic Integers and Conductors 5 1.1 Real Cyclotomic Integers . .5 1.2 Conductor n . .6 2 Defining and Representing Perron Numbers 8 3 Properties of Perron Numbers 9 3.1 Matrices . .9 3.2 Fusion Graphs . 11 3.3 Arithmetic Properties . 15 4 Representing Cyclotomic Integers 16 4.1 Generating Sets . 16 4.2 Length of the Perron Vector . 17 5 Checking if a Cyclotomic Integer is Perron 18 6 Minimality of Perron Numbers 20 6.1 Definition of Minimality . 20 6.2 Relation to Prime Numbers in the Integers . 23 7 Minimality of Quantum Numbers 24 8 Potential Future Applications 30 2 II Algorithms Regarding Perron Numbers and Polynomi- als 31 9 Efficiency and Computational Complexity 31 10 Minimal Perron Vector Algorithms 34 11 Other Computations Related to Perron Numbers 37 12 Perron Research Algorithms Involving Polynomials 38 12.1 Overview of Polynomials in Computer Science . 38 12.2 Polynomial Structures . 40 12.3 Polynomial Operations . 43 12.3.1 Arithmetic within Polynomials . 43 12.3.2 Arithmetic within Expressions and Mods . 57 12.4 Lexing and Parsing . 59 13 Research Summary 65 14 Conclusion 66 3 Abstract Although they are linked to many areas of mathematics, Perron numbers remain a field into which relatively little research has been conducted throughout history. Having their roots in matrix mathemat- ics and linear algebra, Perron numbers manifest themselves everywhere from complex patterns in nature to solutions of higher order polynomial equations. This paper discusses our research into cyclotomic Perron numbers and their properties, and is divided into two parts. In Part 1, we introduce the concept of Perron numbers and their properties in algebra and number theory, as well as their relevance to fusion graphs and their spectral radii. Unlike previous research, we represent cyclotomic Perron numbers as vectors of integers, rather than as algebraic expressions or irrational decimals. With this easier to understand vector representation, we describe cyclotomic Perron numbers of a given conductor as points that lie within a \cone" in a multidimensional Cartesian coordinate plane. We then define mini- mal Perron numbers as being points that lie on the boundary of this cone, which can be added together to produce all other Perron num- ber points within the cone. Drawing comparisons to factoring positive integers into prime numbers, we provide new procedures for how to test if a Perron number is minimal. We then detail our findings for various conductors and extrapolate the patterns we discover to explain previously unnoticed properties of cyclotomic Perron numbers. In Part 2, we delve deeper into the algorithms we used in the course of our research and explain their relevance to Perron numbers. We be- gin by expanding our description of our minimal finding algorithms and providing a few general programs for testing if a cyclotomic integer is Perron. We go on to illustrate the importance of large polynomial computations to our research, and discuss some C++ programs we wrote to solve these problems quickly and without making mistakes. We describe the data structures and algorithms involved, including the time and space complexities of each mathematical operation and the grammar used to parse polynomial expressions. We then conclude the paper with a passage on how our discoveries and programs may be use- ful to both future researchers studying Perron numbers and computer scientists writing programs with algebra in mind. 4 Part I The Mathematics of Cyclotomic Perron Numbers 1 Introduction to Cyclotomic Integers and Con- ductors 1.1 Real Cyclotomic Integers Before we can examine Perron numbers, we must define some terms in alge- bra and number theory. We will use these definitions to show where Perron numbers fit into mathematics, how they are categorized, and their special properties. First, we must define algebraic numbers. An algebraic number is any number, real or complex, that can be expressed as a solution of a polynomial equation with rational coefficients. Of these, an algebraic integer is an algebraic number that can be expressed as the root of a polynomial equation with integer coefficients such that the leading coefficient is a 1, meaning the polynomial must be monic. The minimal polynomial of an algebraic integer x is the monic polynomial P (x) with integer coefficients and the least number of factors such that x is a root of P (x). For example, the minimal polynomial of the algebraic integer x = 1 would be x − 1 = 0, and 2 not x − 3x + 2 = (x − 1)(x − 2) = 0, which has anp extra factor. Similarly, 2 the minimal polynomialp of thep algebraic integer x = 2 would be x −2 = 0 and not just x − 2 = 0, as 2 is not an integer coefficient. Additionally, 1 the rational algebraic number x = 2 would not be considered an algebraic 1 integer, as its minimal polynomial is x − 2 = 0, one of whose coefficients is not an integer. A special kind of algebraic integer results from the equation xn − 1 = 0. By the Fundamental Theorem of Algebra, there are n such roots to this equation. Of these, 1 is the only solution that is a real number, along with −1 if n is even; the rest are complex numbers lying on the unit circle in the complex plane. These are known as the complex nth roots of unity. In order to understand them, we must examine Euler's Identity, which extends exponentiation to imaginary exponents. It states that eiθ = cos(θ) + i sin(θ) and yields the interesting equation eiπ + 1 = 0. This means that e2πi = e2πki = 1k = 1 as long as k is an integer. Therefore, the complex 2πki n 2πk 2πk nth roots of unity must all be of the form e = cos( n )+i sin( n ). Here, 5 for the sake of convenience, we will refer to these roots as powers of "n = 2π 2π cos( n ) + i sin( n ). This is because "n is guaranteed to be a primitive nth root of unity, meaning that all other nth roots of unity can be expressed as integer exponents of "n. From here, we are ready to define cyclotomic numbers. A cyclotomic number is a number that can be written as a sum of rational multiples of powers of the complex nth roots of unity. Examples of these would in- 3 3 clude "5 + "5, 4 i, or any rational number. Of the set of cyclotomic numbers, we define cyclotomic integers as cyclotomic numbers that are also al- gebraic integers. In the course of this paper, we will be focusing solely on real cyclotomic integers whose imaginary part is 0; if we refer to cy- clotomic integers, we will assume that they are real unless specified oth- erwise. We note that adding a power of "n to its reciprocal results in k −k 2πk 2πk 2πk 2πk 2πk "n + "n = cos( n ) + i sin( n ) + cos(− n ) + i sin(− n ) = cos( n ) + 2πk 2πk 2πk 2πk i sin( n ) + cos( n ) − i sin( n ) = 2 cos( n ), causing the cosines to com- bine and the sines to cancel each other out, thus removing the imaginary component of the complex number. Indeed, according to Washington (1982), all cyclotomic integers are sums of integer multiples of the complex roots of unity; furthermore, all real cyclotomic integers can be expressed as sums of −1 2π powers of "n + "n = 2 cos( n ). This is an extremely important property of cyclotomic integers, and we will make good use of it when constructing and categorizing Perron numbers later on. 1.2 Conductor n In the course of our research, we came up with a method for categorizing cyclotomic integers into similar groups known as conductors. In this section, we will discuss what a conductor is and how it can be used to describe the properties of cyclotomic integers as well as Perron numbers. From previous sections, we can surmise that the easiest way to group cyclotomic integers would be based on the nth roots of unity of which they −16 5 are comprised. For instance, "7 + "7 = "7 + "7 would be of conductor 7, 3 π 2π "3 + "9 = "9 + "9 would be of conductor 9, and 2 cos( 5 ) = 2 cos( 10 ) = "10 + −1 2 −2 "10 = −"5 − "5 would be of conductor 5. Indeed, upon close examination, our method would seem to work well for even conductors. For odd ones, however, we run into some difficulties. For instance, let us look at the complex fifth roots of unity in Figure 1. Because they are generated from 2π the sines and cosines of the angle 5 , which is one fifth of the unit circle, they form a regular pentagon in the complex plane (outlined in blue), which is a pattern we can extrapolate to regular n-gons for all complex nth roots 6 of unity.
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