Théorème de Roth

En mathématiques, le théorème de Roth, ou théorème de Thue-Siegel-Roth, est un énoncé de théorie des nombres, concernant plus particulièrement l'approximation diophantienne.

Le résultat est le suivant[1] :

Pour tout nombre irrationnel algébrique α et pour tout ε > 0, l'inéquation d'inconnues q > 0 et p entiers :

n'a qu'un nombre fini de solutions (ce n'est plus le cas pour ε = 0, d'après le théorème d'approximation de Dirichlet).

Ou encore, sous les mêmes hypothèses[2],[3] : il existe une constante A > 0 (dépendant de α et ε) telle que

Ceci signifie que la mesure d'irrationalité d'un nombre irrationnel algébrique est égale à 2 et permet, par contraposition, de montrer la transcendance de certains nombres (cependant, le nombre e, qui est transcendant, échappe à ce critère[2] : sa mesure d'irrationalité est égale à 2). Ce théorème est d'ailleurs une généralisation du théorème de Liouville qui avait été historiquement le premier critère de transcendance connu.

Ce résultat a valu à Klaus Roth[4] la médaille Fields en 1958.

Notes et références

  1. ↑ (en) Steven R. Finch, Mathematical Constants, CUP, , 602 p. (ISBN 978-0-521-81805-6, lire en ligne), p. 171-172
  2. ↑ a et b (en) Daniel Duverney, Number Theory : An Elementary Introduction Through Diophantine Problems, World Scientific, coll. « Monographs in Number Theory » (no 4), , 335 p. (ISBN 978-981-4307-46-8, lire en ligne), p. 147
  3. ↑ (en) Yann Bugeaud, Approximation by Algebraic Numbers, CUP, , 292 p. (ISBN 978-0-521-82329-6, lire en ligne), p. 28
  4. ↑ (en) K. F. Roth, « Rational approximations to algebraic numbers », Mathematika, vol. 2, no 1,‎ , p. 1-20 (DOI 10.1112/S0025579300000644) et "Corrigendum", p. 168, DOI 10.1112/S0025579300000826.

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