Quadratic Forms on F q [ T ]

Author: Car M.

Source: Journal of Number Theory, Volume 61, Number 1, November 1996 , pp. 145-180(36)

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Abstract:

In this paper, we study the number of representations of polynomials of the ring F q [ T ] by diagonal quadratic forms A_1 Y^2_1+\cdots +A_s Y^2_s ,\eqno (Q) 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 >=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 {\bf F}_q[T](\sqrt{-D}) satisfying the most restrictive degree conditions.

Language: English

Document Type: Research article

Affiliations: Laboratoire de Mathematiques, Faculte des Sciences de St-Jerome, Avenue Escadrille Normandie Niemen, Marseille Cedex 20, 13397, France

Publication date: 1996-11-01