  These notes form part of a joint research project on the logic of fields with
many valuations, connected by a product formula. We define such structures and
name them {\em globally valued fields} (GVFs).
  This text aims primarily at a proof that {\em the canonical GVF structure on
$k(t)^{alg}$ is existentially closed}. This can be read as saying that a
variety {\em with a distinguished curve class} is a good approximation for a
formula in the language of GVFs, in the same way that a variety is close to a
formula for the theory ACF of algebraically closed fields.
