Zeros of a Pair of Quadratic Forms Defined over a Finite Field

LetNbe the number of affine zeros of a pair of quadratic forms inn+1 variables defined over a finite fieldFq. We give upper and lower bounds forNand show that these bounds are optimal. One result states that ifn+1≥10 and every quadratic form in the pencil has order at least three, then N−qn−1