Positive values of non-homogeneous quadratic forms of type (1,4): A conjecture of Bambah, Dumir and Hans-Gill

Let $Q(x_1,\cdots ,x_n)$ be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature $\sigma =r-s$ and determinant $D\mathrm{\ne }0$. Let ${\mathrm{\Gamma }}_{r,n-r}$ denote the infimum of all numbers $\mathrm{\Gamma }$ such that for any real numbers $c_1,c_2,\cdots ,c_n$ there exist integers $x_1,x_2,\cdots ,x_n$ satisfying $$0<Q(x_1+c_1,x_2+c_2,\cdots ,x_n+c_n)\le (\mathrm{\Gamma }\mathrm{\mid }D\mathrm{\mid }{)}^{1\mathrm{/}n}\mathrm{.}$$ All the values of ${\mathrm{\Gamma }}_{r,n-r}$ are known except for ${\mathrm{\Gamma }}_{1,4}$. Earlier it was shown that $8\le {\mathrm{\Gamma }}_{1,4}<12$. It is conjectured that ${\mathrm{\Gamma }}_{1,4}=8$. Here we shall prove that ${\mathrm{\Gamma }}_{1,4}=8$, when (i) $c_2\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}1)$, (ii) $c_2\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}1)$, $a\ge \frac{1}{2}$, where $a$ is minima of positive definite ternary quadratic forms with determinant $4\mathrm{\mid }D\mathrm{\mid }$, and (iii) in some cases of $c_2\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}1)$, $a<\frac{1}{2}$. We also obtain six critical forms for which the constant 8 is attained. In the remaining cases we prove that ${\mathrm{\Gamma }}_{1,4}<\frac{32}{3}$.