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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 firstpolynomial bound for this problem. In particular, the bounds coming from thetheory of fewnomials are exponential in $t$, and count only nondegeneratesolutions. More generally, we show that if the set of solutions is infinite, itstill has at most $O(d^3t+d^2t^3)$ connected components. By contrast, thefollowing 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 inalgebraic complexity theory and upper bounds on the number of real roots of"sparse like" polynomials.

On the intersection of a sparse curve and a low-degree curve: A polynomial version of the lost theorem