scholarly journals The Mahler measure and its areal analog for totally positive algebraic integers

2015 ◽  
Vol 151 ◽  
pp. 211-222
Author(s):  
V. Flammang
Keyword(s):  
Mahler Measure ◽  
Algebraic Integers ◽  
Totally Positive

scholarly journals On the Zhang–Zagier measure

2018 ◽  
Vol 14 (10) ◽  
pp. 2663-2671
Author(s):  
V. Flammang
Keyword(s):  
Lower Bound ◽  
Mahler Measure ◽  
Algebraic Integers ◽  
Totally Positive ◽  
The Absolute

We first improve the known lower bound for the absolute Zhang–Zagier measure in the general case. Then we restrict our study to totally positive algebraic integers. In this case, we are able to find six points for the related spectrum. At last, we give inequalities involving the Zhang–Zagier measure, the Mahler measure and the length of such integers.

scholarly journals THE TOTAL DISTANCE FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

2014 ◽  
Vol 90 (3) ◽  
pp. 391-403
Author(s):  
V. FLAMMANG
Keyword(s):  
Recursive Algorithm ◽  
Mahler Measure ◽  
Algebraic Integers ◽  
Numerical Examples ◽  
Auxiliary Functions ◽  
Total Distance ◽  
Totally Positive ◽  
Elementary Inequality

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}P(x)$ be a polynomial of degree $d$ with zeros $\alpha _1, \ldots, \alpha _d$. Stulov and Yang [‘An elementary inequality about the Mahler measure’, Involve6(4) (2013), 393–397] defined the total distance of$P$ as ${\rm td}(P)=\sum _{i=1}^{d} | | \alpha _i| -1|$. In this paper, using the method of explicit auxiliary functions, we study the spectrum of the total distance for totally positive algebraic integers and find its five smallest points. Moreover, for totally positive algebraic integers, we establish inequalities comparing the total distance with two standard measures and also the trace. We give numerical examples to show that our bounds are quite good. The polynomials involved in the auxiliary functions are found by a recursive algorithm.

Reciprocal Algebraic Integers Whose Mahler Measures are Non-Reciprocal

1987 ◽  
Vol 30 (1) ◽  
pp. 3-8 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Unit Circle ◽  
Algebraic Integer ◽  
Mahler Measure ◽  
Algebraic Integers

AbstractThe Mahler measure M (α) of an algebraic integer α is the product of the moduli of the conjugates of α which lie outside the unit circle. A number α is reciprocal if α- 1 is a conjugate of α. We give two constructions of reciprocal a for which M (α) is non-reciprocal producing examples of any degree n of the form 2h with h odd and h ≥ 3, or else of the form with s ≥ 2. We give explicit examples of degrees 10, 14 and 20.

Totally positive algebraic integers with small trace

2021 ◽  
pp. 1
Author(s):  
Cong Wang ◽  
Jie Wu ◽  
Qiang Wu
Keyword(s):  
Algebraic Integers ◽  
Totally Positive

scholarly journals The trace of totally positive algebraic integers

2005 ◽  
Vol 75 (253) ◽  
pp. 385-394 ◽  
Author(s):  
Julián Aguirre ◽  
Mikel Bilbao ◽  
Juan Carlos Peral
Keyword(s):  
Algebraic Integers ◽  
Totally Positive

The Trace of Totally Positive and Real Algebraic Integers

1945 ◽  
Vol 46 (2) ◽  
pp. 302 ◽  
Author(s):  
Carl Ludwig Siegel
Keyword(s):  
Algebraic Integers ◽  
Totally Positive
Sign in / Sign up

Export Citation Format

Share Document