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

Fecha:	2007
Resumen:	The classical regulator R of a number field K is given [F1] by a rapidly convergent series of the form 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, ∞) → R is defined as 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/ $) , which is useful for certain ranges of D.
Derechos:	Tots els drets reservats.
Lengua:	Anglès
Documento:	Article ; recerca ; Versió publicada
Materia:	Regulator ; Total positivity
Publicado en:	Publicacions matemàtiques, Vol. Extra (2007) , p. 119-130, ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/69984
DOI: 10.5565/PUBLMAT_PJTN05_05

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