We construct certain raising operators of row type for Macdonald's symmetric polynomials by an interpolation method.

The Hermitian function field H= K(x,y) is defined by the equation yq+y=xq +1 (q being a power of the characteristic of K). Over K=${\mirrored F}$q2 it is a maximal function field; i.e. the number N(H) of ${\mirrored F}$q2-rational places attains the Hasse–Weil upper bound N(H)=q2+1+2g(H)·q. All subfields K[subnE ] E⊆H are also maximal. In this paper we construct a large number of nonrational subfields E⊆H, by considering the fixed fields H$^{\mathcal G}$ under certain groups ${\mathcal G}$ of automorphisms of H/K. Thus we obtain many integers g[ges ]0 that occur as the genus of some maximal function field over ${\mirrored F}$q2.

Moduli spaces of pointed curves with some level structure are studied. We prove that for so-called geometric level structures, the levels encountered in the boundary are smooth if the ambient variety is smooth, and in some cases we can describe them explicitly. The smoothness implies that the moduli space of pointed curves (over any field) admits a smooth finite Galois cover. Finally, we prove that some of these moduli spaces are simply connected.

We will give explicit bounds for the number of solutions of polynomial-exponential equations. In contrast to earlier work, the bounds are independent of the coefficients of the equations, and they are of only single exponential growth in the number of coefficients.