Approximation of Algebraic Numbers

Abstract

Let

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaHvpGzdaqadaWdaeaapeGaamOEaaGaayjkaiaawMcaaiabg2da % 9iabew9aMnaabmaapaqaa8qacaWG6bWdamaaBaaaleaapeGaaGymaa % WdaeqaaOWdbiaacYcacqGHMacVcaGGSaGaamOEa8aadaWgaaWcbaWd % biaad2gaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpdaaeWbWdae % aapeGaeS47IW0aaabCa8aabaWdbiaadgeapaWaaSbaaSqaa8qacaWG % 2bWdamaaBaaameaapeGaaGymaaWdaeqaaSWdbiaacYcacqGHMacVca % GGSaGaamODa8aadaWgaaadbaWdbiaad2gaa8aabeaaaSqabaGcpeGa % amOEa8aadaqhaaWcbaWdbiaaigdaa8aabaWdbiaadAhapaWaaSbaaW % qaa8qacaaIXaaapaqabaaaaOWdbiabl+UimjaadQhapaWaa0baaSqa % a8qacaWGTbaapaqaa8qacaWG2bWdamaaBaaameaapeGaamyBaaWdae % qaaaaaaSqaa8qacaWG2bWdamaaBaaameaapeGaamyBaaWdaeqaaSWd % biabg2da9iaaicdaa8aabaWdbiaad6gapaWaaSbaaWqaa8qacaWGTb % aapaqabaaan8qacqGHris5aOGaeyicI4mal8aabaWdbiaadAhapaWa % aSbaaWqaa8qacaaIXaaapaqabaWcpeGaeyypa0JaaGimaaWdaeaape % GaamOBa8aadaWgaaadbaWdbiaaigdaa8aabeaaa0WdbiabggHiLdGc % caWGdbWaamWaa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaaIXaaapa % qabaGcpeGaaiilaiabgAci8kaacYcacaWG6bWdamaaBaaaleaapeGa % amyBaaWdaeqaaaGcpeGaay5waiaaw2faaaaa!7A81! $$\phi \left( z \right) = \phi \left( {{z_1}, \ldots ,{z_m}} \right) = \sum\limits_{{v_1} = 0}^{{n_1}} { \cdots \sum\limits_{{v_m} = 0}^{{n_m}} {{A_{{v_1}, \ldots ,{v_m}}}z_1^{{v_1}} \cdots z_m^{{v_m}}} \in } C\left[ {{z_1}, \ldots ,{z_m}} \right]$$