×

Passage d’une mesure d’approximation à une mesure de transcendance. (Passage from an approximation measure to a transcendence measure). (French) Zbl 0763.11032

An approximation measure for a transcendental number \(\omega\) is a lower bound for \(|\omega-\alpha|\), when \(\alpha\) is an algebraic number; such an estimate depends on the degree of \(\alpha\) as well as, say, the (usual) height of \(\alpha\). A transcendence measure for \(\omega\) is a lower bound for \(| P(\omega)|\), when \(P\in\mathbb{Z}[X]\) is a non-zero polynomial, in terms of the degree \(D\) of \(P\) and of the (usual) height \(H\) of \(P\). Most often, transcendence proofs provide an approximation measure, from which one deduces a transcendence measure. However this deduction, so far, introduced an error term involving \(D(D+\log H)\). In this note, using Chudnovsky’s semi-resultant, the authors replace the error term by \(D(\log D+\log H)\). This estimate is useful when sharp estimates (in term of the degree) are considered; see for instance G. Diaz [Approximations diophantiennes et nombres transcendants, C. R. Colloq., Luminy 1990, 105-121 (1992; Zbl 0758.11031)].

MSC:

11J82 Measures of irrationality and of transcendence

Citations:

Zbl 0758.11031
PDFBibTeX XMLCite