Quadratic forms on F-q[T]

In this paper, we study the number of representations of polynomials of the ring F-q[T] by diagonal quadratic forms A(1) Y-1(2) + ... + A(s) Y-s(2), where A(1), ..., A(s) are given polynomials and Y-1, ..., Y-s are polynomials subject to satisfying the most restrictive degree conditions. When A(1), ..., A(s) are pairwise coprime, and s greater than or equal to 5, we use the ordinary circle method; when A(1), ..., A(4) are pairwise coprime we adapt Kloosterman's method to the polynomial case and we get an asymptotic estimate for the number R(A(1), ..., A(s); M) of representations of a polynomial M as a sum(Q). We also deal with the particular case s = 4, A(1) = A(2) = D, A(3) = A(4) = 1, where D is a square-free polynomial. In this particular case, the number R(A(1), ..., A(4); M) is the number of representations of M as a sum of two norms of elements of the quadratic extension F-q[T](root-D) satisfying the most restrictive degree conditions. (C) 1996 Academic Press, Inc.