Criteria of algebraic independence of complex numbers over a field of finite transcendence type (II)

Abstract

In the present paper we establish a criterion of algebraic independence of complex numbersθ ₁, ...,θ _n over a field$\mathbb{K}$ ⊂ ℂ of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients, which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point (ω ₁, ...,ω _q ,θ ₁, ...,θ _n ), where {ω ₁, ...,ω _q } is a transcendence basis of$\mathbb{K}$ over ℚ.