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Regulators and total positivity
Friedman, Eduardo (Universidad de Chile. Departamento de Matemática)

Data: 2007
Resum: The classical regulator R of a number field K is given [F1] by a rapidly convergent series of the form R w = X∞ m=1 amg(m2 /|D|), where w is the number of roots of unity in K, D is the discriminant of K, am counts certain integral ideals in K of absolute norm m, and g : (0, ∞) 7→ R is defined as g(x) := 1 2 r1 4πi Z 2+i∞ 2−i∞ (4r2 π [K:Q]x) −s/2 (2s − 1)Γ` s 2 ´r1 Γ(s) r2 ds, r1 and r2 being, respectively, the number of real and complex places of K. If the unit group of K is infinite, it is known that g(x) tends to −∞ as x → 0 +, and that g(x) is positive and vanishes exponentially fast for large x. Using classical results from the theory of total positivity we prove that g has the simplest possible behavior compatible with these asymptotic data. Namely, g(x) has a unique zero in (0, ∞), and the same holds for each derivative of g. This leads to a new lower bound for the regulator R > w g(1/|D|), which is useful for certain ranges of D.
Drets: Tots els drets reservats
Llengua: Anglès
Document: article ; recerca ; publishedVersion
Matèria: Regulator ; Total positivity
Publicat a: Publicacions matemàtiques, Vol. Extra (2007) , p. 119-130, ISSN 2014-4350

Adreça original:
DOI: 10.5565/PUBLMAT_PJTN05_05
DOI: 10.5565/69984

12 p, 157.2 KB

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