Robert M. Guralnick, Joel E. Rosenberg, and Michael E. Zieve:
A new family of exceptional polynomials in characteristic two,
Annals of Math. 172 (2010), 1361–1390.

(Both the published version and the arXiv version are available online.)

A polynomial g over Fq is called exceptional if the map ag(a) induces a bijection on Fqn for infinitely many n. The composition of two polynomials is exceptional if and only if both polynomials are exceptional, so it suffices to study the indecomposable exceptional polynomials. Letting  p denote the characteristic of  FqFried, Guralnick and Saxl showed that any indecomposable exceptional polynomial has degree either prime or a power of  p, except perhaps when  p≤3  in which case they could not rule out the possibility that the degree is  pr(pr-1)/2  with r > 1 odd. In our previous paper, we determined all exceptional polynomials in this last situation, except for one ramification configuration; these polynomials were twists of exceptional polynomials discovered previously by Müller, Cohen–Matthews, and Lenstra–Zieve. In this paper we complete the classification of indecomposable exceptional polynomials of non-prime power degree, by addressing this final ramification configuration. It turns out that this yields a new family of indecomposable exceptional polynomials, which includes polynomials of degree 2r-1(2r-1) over F2s whenever r > 1 is odd and s > 1 is coprime to r.

The strategy of our proof is to identify the curves C which can occur as the Galois closure of the cover g : P1P1 for a polynomial g satisfying the required properties. It turns out that any such C is geometrically isomorphic to the smooth plane curve yq+1+zq+1=T(yz)+c, where Q=2r with r > 1 odd, and where T(X)=Xq/2+Xq/4+...+X. This family of curves is of independent interest, since each curve in the family has ordinary Jacobian and has automorphism group significantly larger than its genus. A key step in our proof is the computation of the automorphism groups of curves of the form vq+v=h(w), with h varying over a two-parameter family of rational functions. Our method for this is rather general, and applies to many families of rational functions h.

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