The zeros of a quadratic form at square-free points (Q708250)

The present paper establishes asymptotic formulae for the number of \textit{squarefree} solutions to \(F(\mathbf{x}) = 0\) where \(F\) is a non-singular, indefinite quadratic form in 3 or 4 variables and the variables are restricted to a large box \(x_j \asymp P\). Some technical conditions on the subdeterminants of the matrix of \(F\) are imposed, and the count is weighted by a smoothing function. The error term in the asymptotic formula saves a power of \(\log\log P\). This paper is a continuation of [Acta Arith. 124, No. 2, 101--137 (2006; Zbl 1152.11043)] and an application of \textit{D. R. Heath-Brown}'s new form of the circle method [J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)].