  Consider a system of two polynomial equations in two variables:
$$F(X,Y)=G(X,Y)=0$$ where $F \in \rr[X,Y]$ has degree $d \geq 1$ and $G \in
\rr[X,Y]$ has $t$ monomials. We show that the system has only $O(d^3t+d^2t^3)$
real solutions when it has a finite number of real solutions. This is the first
polynomial bound for this problem. In particular, the bounds coming from the
theory of fewnomials are exponential in $t$, and count only nondegenerate
solutions. More generally, we show that if the set of solutions is infinite, it
still has at most $O(d^3t+d^2t^3)$ connected components. By contrast, the
following question seems to be open: if $F$ and $G$ have at most $t$ monomials,
is the number of (nondegenerate) solutions polynomial in $t$? The authors'
interest for these problems was sparked by connections between lower bounds in
algebraic complexity theory and upper bounds on the number of real roots of
"sparse like" polynomials.
