Positive values of non-homogeneous quadratic forms of type (1,4): A conjecture of Bambah, Dumir and Hans-Gill (2402.18939v1)

Abstract: 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 $D\neq 0$. Let $\Gamma_{r,n-r}$ denote the infimum of all numbers $\Gamma$ such that for any real numbers $c_1, c_2 ,\cdots, c_n$ there exist integers $x_1, x_ 2,\cdots, x_n$ satisfying $$0< Q(x_1+c_1,x_2+c_2,\cdots,x_n+c_n)\leq (\Gamma |D|)^{1/n}.$$ All the values of $\Gamma_{r,n-r}$ are known except for $\Gamma_{1,4}$. Earlier it was shown that $8\leq \Gamma_{1,4}<12$. It is conjectured that $\Gamma_{1,4}=8$. Here we shall prove that $\Gamma_{1,4}=8$, when (i) $c_2 \not \equiv 0 \pmod 1$, (ii) $c_2 \equiv 0 \pmod 1$, $a\geq \frac{1}{2}$, where $a$ is minima of positive definite ternary quadratic forms with determinant $4|D|$, and (iii) in some cases of $c_2 \equiv 0 \pmod 1$, $a< \frac{1}{2}$. We also obtain six critical forms for which the constant 8 is attained. In the remaining cases we prove that $\Gamma_{1,4}< \frac{32}{3}$.