Simultaneous Approximation and Algebraic Independence

We establish several new measures of simultaneous algebraic approximations for families of complex numbers related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the field has transcendance degree 2 over . This approach which is ultimately based on the technique of interpolation determinants provides an alternative to Gelfonds transcendence criterion. We also formulate a conjecture about simultaneous algebraic approximation which would yield higher transcendance degrees from these measures.