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Quadratic forms on \(\mathbb{F}_ q[T]\). (English) Zbl 0870.11076

Let \(q\) be a power of an odd prime number and \(\mathbb{F}_q\) denote the finite field with \(q\) elements. In this paper the author studies the number of representations of certain polynomials \(M\in\mathbb{F}_q[T]\) by diagonal quadratic forms. That is, \[ M=A_1M_1^2+\cdots+ A_sM_s^2,\tag{1} \] where \(A_1,\dots,A_s\) are fixed polynomials and \(M_1,\dots,M_s\) are any polynomials satisfying \[ \deg M_j\leq m_j \quad\text{if}\quad \deg M-\deg A_j\in\{2m_j;2m_j-1\},\tag{2} \] for \(j=1,\dots,s\).
By the circle method, the author proves that for \(s\geq 5\), if \(A_1,\dots, A_s\) are pairwise coprime and \(\deg M\) is large enough such that the conditions in (2) are satisfied, then the number of representations in (1) is \[ \gg q^{(s-1)(\deg M)/2}, \] where the implied constant in the \(\gg\) depends on \(q\) and \(A_1,\dots, A_s\) only. If \(s\) is fixed to be 4, then by Kloosterman’s method the author proves that the number of representations is \[ \gg q^{\deg M}(\log(\deg M))^{-1}. \] Some particular cases are also discussed in the paper.

MSC:

11T55 Arithmetic theory of polynomial rings over finite fields
11P99 Additive number theory; partitions
11T06 Polynomials over finite fields
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