Variation of a Theme of Landau-Shanks in Positive Characteristic

Let A := F q [t] be a polynomial ring over a finite field F q of odd characteristic and let D ∈ A be a square-free polynomial. Denote by N D (n, q) the number of polynomials f in A of degree n which may be represented in the form u·f = A2–DB2 for some A, B ∈ A and u ∈ F q × , and by BD(n, q) the number of polynomials in A of degree n which can be represented by a primitive quadratic form of a given discriminant D ∈ A, not necessary square-free. If the class number of the maximal order of F q (t, √D) is one, then we give very precise asymptotic formulas for N D (n, q). Moreover, we also give very precise asymptotic formulas for BD(n, q).