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 = A 2 −DB 2 for some A, B ∈ A and u ∈ F × q , and by B D (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 B D (n, q).
We give an asymptotic formula for the proportion of polynomials of a given degree over a finite field which do not have odd degree irreducible factors. In the case of odd characteristic, this leads to an asymptotic formula for certain weighted partition function which describes the major proportion of the fundamental discriminants where the "negative" Pell equation cannot be solved. We also extend the results to counting positive divisors over an arbitrary global function field.
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