different method by Zahl. The implicit constant in the bound is not the same in the complex numbers: in Tóth's proof the constant can be taken to be ; the constant is not explicit in Zahl's proof. When the point set is a Cartesian product, Solymosi and Tardos show that the Szemerédi-Trotter bound holds using a much simpler argument. In finite fields Let be a field. A Szemerédi-Trotter bound is impossible in general due to the following example, stated here in : let be the set of all points and let be the set of all lines in the plane. Since each line contains points, there are incidences. On the other hand, a Szemerédi-Trotter bound would give incidences. This example shows that the trivial, combinatorial incidence bound is tight. Bourgain, Katz and Tao show that if this example is excluded, then an incidence bound that is an improvement on the trivial bound can be attained. Incidence bounds over finite fields are of two types: (i) when at least one of the set of points or lines is `large' in terms of the characteristic of the field; (ii) both the set of points and the set of lines are `small' in terms of the characteristic. Large set incidence bounds Let be an odd prime power. Then Vinh showed that the number of incidences between points and lines in is at most Note that there is no implicit constant in this bound. Small set incidence bounds Let be a field of characteristic . Stevens and de Zeeuw show that the number of incidences between points and lines in is under the condition in positive characteristic. (In a field of characteristic zero, this condition is not necessary.) This bound is better than the trivial incidence estimate when . If the point set is a Cartesian Product, then they show an improved incidence bound: let be a finite set of points with and let be a set of lines in the plane. Suppose that and in positive characteristic that . Then the number of incidences between and is This bound is optimal. Note that by point-line duality in the plane, this incidence bound can be rephrased for an arbitrary point set and a set of lines having a Cartesian product structure. In both the reals and arbitrary fields, Rudnev and Shkredov show an incidence bound for when both the point set and the line