Ramanujan Expansion over Algebraic Integer Rings

2021 ◽  
Vol 11 (04) ◽  
pp. 442-453
Author(s):  
旭瑞 刘
Keyword(s):  
Algebraic Integer ◽  
Integer Rings

Arithmetic definability by formulas with two quantifiers

1992 ◽  
Vol 57 (1) ◽  
pp. 1-11
Author(s):  
Shih Ping Tung
Keyword(s):  
Algebraic Number ◽  
Necessary Conditions ◽  
Algebraic Integer ◽  
Number Fields ◽  
Universal Quantifier ◽  
Existential Quantifier ◽  
Algebraic Number Fields ◽  
Integer Rings

AbstractWe give necessary conditions for a set to be definable by a formula with a universal quantifier and an existential quantifier over algebraic integer rings or algebraic number fields. From these necessary conditions we obtain some undefinability results. For example, N is not definable by such a formula over Z. This extends a previous result of R. M. Robinson.

scholarly journals A sequence of algebraic integer relation numbers which converges to 4

2021 ◽  
Vol 294 ◽  
pp. 107665
Author(s):  
Wonyong Jang ◽  
KyeongRo Kim
Keyword(s):  
Algebraic Integer

scholarly journals Non-negative integral matrices with given spectral radius and controlled dimension

2021 ◽  
pp. 1-24
Author(s):  
MEHDI YAZDI
Keyword(s):  
Spectral Radius ◽  
Positive Real Number ◽  
Algebraic Integer ◽  
Log P ◽  
Irreducible Matrix ◽  
Positive Real ◽  
Shift Of Finite Type ◽  
Primitive Matrix ◽  
Integral Matrices ◽  
Perron Number

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.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (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.

Supplement to The Maximal Modulus of an Algebraic Integer

1985 ◽  
Vol 45 (171) ◽  
pp. S17 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Algebraic Integer ◽  
Maximal Modulus
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