  We begin a generalized study of sum-product type phenomenon in different
fields by considering pairs $P(x,y)$ and $Q(x,y)$ of two variable polynomials
that simultaneously exhibit small symmetric expansion. Our first result is that
such $P(x,y)$ and $Q(x,y)$ over $\mathbb{R}$ and $\mathbb{C}$ have very similar
structure, obtained by employing semi-algebraic geometry/o-minimality. Then
using model-theoretic transfer and basic Galois theory we deduce results for
fields of characteristic $0$ and characteristic $p$ when $p$ is large.
  We obtain as corollaries a generalization of Elekes-R\'onyai type structural
results to arbitrary characteristic 0 fields, and a strengthening of these
classic results in a symmetric case of natural interest. We note a related
bound of $5/4$ in the exponent for the sum-product problem in finite fields of
large characteristic, although a lower bound for this characteristic cannot be
computed from our methods.
