Positive values of non-homogeneous indefinite quadratic forms of type (2,5)
Let \(Q(X)= Q(x_ 1,\dots, x_ n)\) be a real indefinite quadratic form in \(n\) variables of determinant \(D\neq 0\) and type \((r,n-r)\). \textit{H. Blaney} [J. Lond. Math. Soc. 23, 153-160 (1948; Zbl 0031.20404)] proved that there exist real numbers \(\Gamma\), depending only on \(n\) and \(r\), such that for any \(C\in \mathbb{R}^ n\) there exists \(X\in \mathbb{Z}^ n\) such that \[ 0< Q(X+C)\leq (\Gamma| D|)^{1/n}. \tag{1} \] Then the problem arose to determine the minimum \(\Gamma_{r,n-r}\) of all such numbers \(\Gamma\). Further investigations gave the values of \(\Gamma_{r,n-r}\) for various \(r\) and \(n-r\). After the paper by \textit{V. C. Dumir}, \textit{R. J. Hans-Gill} and \textit{A. C. Woods} [J. Number Theory 47, 190-197 (1994; Zbl 0803.11033)] \(\Gamma_{r,n-r}\) was determined for all meaningful pairs \(r\), \(n-r\) with the exception of \(\Gamma_{2,5}\), \(\Gamma_{2,4}\) and \(\Gamma_{1,4}\). They also announced the determination of \(\Gamma_{2,5}\) by V. C. Dumir and R. Sehmi. The present paper confirms this. Namely, it is proved that \(\Gamma_{2,5}= 32\). Thereby the conjecture of Bambah, Dumir and Hans- Gill is confirmed in this case. The paper also determines the forms \(Q(X)\) for which equality in (1) is necessary.
scientific article; zbMATH DE number 638542