Positive values of inhomogeneous indefinite ternary quadratic forms of type \((2,1)\) (Q1918242)

Let \(Q(X)= Q(x_1, \dots, x_n)\) be a real indefinite \(n\)-ary quadratic form of determinant \(D\neq 0\) and of 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 \(0<Q(X+C) \leq (\Gamma |D |)^{1/n}\). Then the problem arose of determining the minimum \(\Gamma_{r,n-r}\) and the \(k\)th successive inhomogeneous minimum \(\Gamma^{(k)}_{r,n-r}\) of all such numbers \(\Gamma\) for the forms \(Q(X)\). Further investigations gave the values of these minima for various \(n,r\) and \(k\). The present paper considers ternary forms of type (2,1) and proves that \(\Gamma^{(2)}_{2,1} = 8/3\). All the critical forms are given. Earlier, for zero ternary forms of the mentioned type the first author [J. Aust. Math. Soc., Ser. A 55, 334-354 (1993; Zbl 0798.11028)] obtained the first four minima.