Determination of all complete mappings of F_{q^2} of the form aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3

For each prime power q, we determine all polynomials over F_{q^2} of the form f(X) := aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3 which induce complete mappings of F_{q^2}, in the sense that each of the functions x --> f(x) and x --> f(x)+x permutes F_{q^2}. This is the first result in the literature which classifies the complete mappings among some class of polynomials with arbitrarily large degree over finite fields of arbitrary characteristic. We also determine all permutation polynomials over F_{q^2} of the form X^{q+2}+bX^q+cX, and all permutations of (F_q)^2 induced by maps of the form (x,y) --> (x^3-exy^2-ax-by, y^3-cx-dy) where either e=0 or 3|q. The latter results add to the small number of results in the literature classifying all permutations induced by maps of prescribed forms.