The Baker—Tijdeman Theorem

Abstract

Let α_l,...,α _r be algebraic numbers, in a number field K. We shall use the notation

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbqec0uyYXwz0rhasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8 % qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9 % q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaqbaO % qaaiaadwhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpciGGSbGaai4B % aiaacEgacqaHXoqydaWgaaWcbaGaamOAaaqabaaaaa!4194! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${u_j} = \log {\alpha _j,}$$

where the log is taken with principal value. We let

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbqec0uyYXwz0rhasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8 % qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9 % q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaqbaO % qaaiaadwfadaWgaaWcbaGaamOAaaqabaGccqGH9aqpciGGSbGaai4B % aiaacEgacaWGibWaaSbaaSqaaiaadUgaaeqaaOWaaeWaaeaacqaHXo % qydaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacaqGHbGaaeOB % aiaabsgacaWGvbGaeyypa0JaciyBaiaacggacaGG4bGaamyvamaaBa % aaleaacaWGQbaabeaakiaacYcaaaa!4F19!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${U_j} = \log {H_k}\left( {{\alpha _j}} \right){\text{and}}U = \max {U_j},$$

We do not assume u _j linearly independent over the integers. In the applications, we shall consider questions of uniformity only with respect to the degree of these algebraic numbers, and so we shall not pay special attention to distinguish the absolute from the relative height.