Aspects of x + y - z = 1 in positive characteristic

In the present thesis we focus on how to solve the equation x+y-z=1 completely in unknowns x,y,z chosen from the group G generated by t and 1-t in the function field F(t), where F is the field with p elements. The general theory is well-understood but there are rather few examples. Here we find at most 243 (so independent of p) families of solutions parametrized by (at most) a pair of prime-power exponents; these correspond to two independent actions of Frobenius. We also indicate how to go further by finding all solutions in the radical of G in the algebraic closure of F(t).