scholarly journals THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

2011 ◽  
Vol 90 (3) ◽  
pp. 341-354 ◽  
Author(s):  
YANHUA LIANG ◽  
QIANG WU
Keyword(s):  
Lower Bounds ◽  
Algebraic Integer ◽  
Algebraic Integers ◽  
Auxiliary Functions ◽  
Salem Numbers ◽  
Totally Positive

AbstractLet α be a totally positive algebraic integer of degree d≥2 and α1=α,α2,…,αd be all its conjugates. We use explicit auxiliary functions to improve the known lower bounds of Sk/d, where Sk=∑ di=1αki and k=1,2,3. These improvements have consequences for the search of Salem numbers with negative traces.

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.

scholarly journals The absolute trace of totally positive algebraic integers

2019 ◽  
Vol 15 (01) ◽  
pp. 173-181
Author(s):  
V. Flammang
Keyword(s):  
Finite Number ◽  
Recursive Algorithm ◽  
Algebraic Integer ◽  
Transfinite Diameter ◽  
Algebraic Integers ◽  
Totally Positive ◽  
The Absolute

Thanks to our recursive algorithm developed in [Trace of totally positive algebraic integers and integer transfinite diameter, Math. Comp. 78(266) (2009) 1119–1125], we prove that, if [Formula: see text] is a totally positive algebraic integer of degree [Formula: see text] with minimum conjugate [Formula: see text] then, with a finite number of explicit exceptions, [Formula: see text]

scholarly journals COMPARISON OF MEASURES OF TOTALLY POSITIVE POLYNOMIALS

2013 ◽  
Vol 89 (2) ◽  
pp. 265-270 ◽  
Author(s):  
V. FLAMMANG
Keyword(s):  
Lower Bounds ◽  
Mahler Measure ◽  
Upper And Lower Bounds ◽  
Positive Polynomials ◽  
Auxiliary Functions ◽  
Integer Coefficients ◽  
Totally Positive

AbstractIn this paper, explicit auxiliary functions are used to get upper and lower bounds for the Mahler measure of monic irreducible totally positive polynomials with integer coefficients. These bounds involve the length and the trace of the polynomial.

On ℤ-Modules of Algebraic Integers

2009 ◽  
Vol 61 (2) ◽  
pp. 264-281 ◽  
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.

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.

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
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