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

A polynomial g over F_q is called exceptional if the map a→g(a) induces a bijection on F_qⁿ 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  F_q,  Fried, 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  p^r(p^r-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 2^r-1(2^r-1) over F_2^s 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 : P¹→P¹ for a polynomial g satisfying the required properties. It turns out that any such C is geometrically isomorphic to the smooth plane curve y^q+1+z^q+1=T(yz)+c, where Q=2^r with r > 1 odd, and where T(X)=X^q/2+X^q/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 v^q+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.

Michael Zieve:  home page   publication list