 Incidences between points and polynomial graphs

For $m$ distinct real univariate polynomials of degree at most $d$ and $n$ points with distinct first coordinates, the number of incidences between the points and the polynomial graphs isTwo polynomial graphs meet at most $d$ times, so the graph formed from consecutive incidences has $O(d{m}^2)$ crossings; the Crossing lemma supplies the lower bound.