Abstract
In this paper, we study the number of representations of polynomials of the ringFq[T] by diagonal quadratic forms[formula]whereA1, …, Asare given polynomials andY1, …, Ysare polynomials subject to satisfying the most restrictive degree conditions. WhenA1, …, Asare pairwise coprime, ands⩾5, we use the ordinary circle method; whenA1, …, A4are pairwise coprime we adapt Kloosterman's method to the polynomial case and we get an asymptotic estimate for the number R(A1, …, As; M) of representations of a polynomialMas a sum(Q). We also deal with the particular cases=4,A1=A2=D,A3=A4=1, whereDis a square-free polynomial. In this particular case, the number R(A1, …, A4; M) is the number of representations ofMas a sum of two norms of elements of the quadratic extension[formula]satisfying the most restrictive degree conditions.
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