Let G be a Sylow p-subgroup of the unitary groups GU (3, q 2 ), GU (4, q 2 ), the sympletic group Sp(4, q) and, for q odd, the orthogonal group O + (4, q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.

The defect numbers of the generalized Hilbert and Carleman boundary value problems with a direct or an inverse linear fractional Carleman shift of order 2 (α (α (t)) ≡ t) on the unit circle are computed. The approach followed consists of the reduction of the mentioned problems to singular integral equations with linear fractional Carleman shift and of the factorization of Hermitian matrix functions with negative determinant. Mathematics Subject Classification (2000). Primary 47G10; Secondary 47A68.

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this paper we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X. Corollary 1.2. Let R be the subalgebra of F[V ] G , generated by G-orbit products of variables and Steenrod images of the form h defining X. Then F[V ] G is equal to the integral closure of R in its fraction field. Proof. By Theorem 1.1, R and F[V ] G have the same Quotient field, say K. It follows from [3] Theorem 4.0.3. pg. 60, that R contains a homogeneous system of parameters of F[V ] G , which is therefore integral over R. Let f ∈ K be integral over R, then f is integral over F[V ] G and therefore contained in the normal ring F[V ] G .

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