Extending the PSLQ Algorithm to Algebraic Integer Relations

Abstract A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

Download Full-text

Algebraic Properties of Chromatic Roots

The Electronic Journal of Combinatorics ◽

10.37236/6578 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 1

Author(s):

Peter J. Cameron ◽

Kerri Morgan

Keyword(s):

Algebraic Integer ◽

Tutte Polynomial ◽

Chromatic Polynomial ◽

Galois Groups ◽

Algebraic Numbers ◽

Isaac Newton ◽

Isaac Newton Institute ◽

Chromatic Root ◽

The Subject ◽

Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

Download Full-text

Supplement to The Maximal Modulus of an Algebraic Integer

Mathematics of Computation ◽

10.2307/2008075 ◽

1985 ◽

Vol 45 (171) ◽

pp. S17 ◽

Cited By ~ 1

Author(s):

David W. Boyd

Keyword(s):

Algebraic Integer ◽

Maximal Modulus

Download Full-text

On ℤ-Modules of Algebraic Integers

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-013-9 ◽

2009 ◽

Vol 61 (2) ◽

pp. 264-281 ◽

Cited By ~ 2

Author(s):

J. P. Bell ◽

K. G. Hare

Keyword(s):

Algebraic Integer ◽

Full Rank ◽

Pisot Number ◽

List Type ◽

Algebraic Integers ◽

Pisot Numbers ◽

Cyclotomic Numbers ◽

Special Cases ◽

Fixed Order ◽

Fixed Positive Integer

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

Download Full-text

An Extension of Fermat's Theorem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1967-036-8 ◽

1967 ◽

Vol 10 (3) ◽

pp. 383-386

Author(s):

Chinthayamma ◽

J. M. Gandhi

Keyword(s):

Algebraic Integer ◽

Image Position

In [1[ Trypanis has proved the following extension of Fermat's theorem: If a is any integer, paprime, , then1This result is to be understood in the sense that a(p-1)/pn-1=p1/pnα where α is an algebraic integer.

Download Full-text

Divisor Sums of Generalised Exponential Polynomials

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-005-5 ◽

1996 ◽

Vol 39 (1) ◽

pp. 35-46 ◽

Cited By ~ 1

Author(s):

G. R. Everest ◽

I. E. Shparlinski

Keyword(s):

Prime Ideal ◽

Algebraic Integer ◽

Linear Recurrence ◽

Exponential Polynomial ◽

Exponential Polynomials ◽

Recurrence Sequence ◽

Special Cases ◽

Divisor Sums ◽

Recurrence Sequences

AbstractA study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

Download Full-text