Gelfond’s method for algebraic independence

Abstract

This paper extends Gelfond’s method for algebraic independence to fields K K with transcendence type ⩽ τ \leqslant \tau . The main results show that the elements of a transcendence basis for K K and at least two more numbers from a prescribed set are algebraically independent over Q Q . The theorems have a common hypothesis: { α 1 , … , α M } , { β 1 , … , β N } \{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\} are sets of complex numbers, each of which is Q Q -linearly independent. THEOREM A. If ( 2 τ − 1 ) > M N (2\tau - 1) > MN , then at least two of the numbers α i , β j , exp ⁡ ( α i β j ) , 1 ⩽ i ⩽ M , 1 ⩽ j ⩽ N {\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N , are algebraically dependent over K K . THEOREM B. If 2 τ ( M + N ) ⩽ M N + M 2\tau (M + N) \leqslant MN + M , then at least two of the numbers α i , exp ⁡ ( α i , β j ) , 1 ⩽ i ⩽ M , 1 ⩽ j ⩽ N {\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N , are algebraically dependent over K K . THEOREM C. If 2 τ ( M + N ) ⩽ M N 2\tau (M + N) \leqslant MN , then at least two of the numbers 1 ⩽ i ⩽ M , 1 ⩽ j ⩽ N 1 \leqslant i \leqslant M,1 \leqslant j \leqslant N , are algebraically dependent over K K .