Criteria of Algebraic Independence of Complex Numbers over a Field of Finite Transcendence Type (II)

In the present paper we establish a criterion of algebraic independence of complex numbers .., ..., .n, over a field X . C of finite transcendence type using a sequence of nonzero polynomials in sev eral variables with integral coefficients.which satisfy simultaneously certain upper and lower estimates in dif ferent orders of magnitude at the point (.., ..., .q, .., ..., .n) where (.., ..., .q) is a transcendence basis of K over Q.