A note on some elementary measures of algebraic independence

Abstract

We investigate the algebraic independence of some numbers associated with elliptic functions when one of the numbers is a "Liouville-type" number. Suppose ℘ ( z ) \wp (z) is a Weierstrass elliptic function with algebraic invariants and β \beta is an algebraic number, not belonging to the field of multiplications for ℘ ( z ) \wp (z) . We establish the algebraic independence of ℘ ( u ) \wp (u) and ℘ ( β u ) \wp (\beta u) (respectively, of u u and ℘ ( β u ) \wp (\beta u) ) when ℘ ( u ) \wp (u) (respectively, u u ) is a "Liouville-type" number. We also give quantitative versions of these results.