Let $M$ be a real symmetric (non-zero) matrix. Consider the quadratic form, \begin{eqnarray} Q(\vec{t})&\equiv&\vec{t}^{\,T} M \,\vec{t} \\ &=& \begin{pmatrix} t_0 ~ \ldots ~ t_N \end{pmatrix} \begin{pmatrix} M_{00} & \ldots & M_{0N} \\ \vdots & \ddots & \vdots \\ M_{N0} & \ldots & M_{NN} \end{pmatrix} \begin{pmatrix} t_0 \\ \vdots \\ t_N \end{pmatrix} \;, \end{eqnarray} where the $t_i$ are all real. My question is as follows.
Is it possible to determine necessary and sufficient conditions on $M$ for there to exist a solution $\vec{t}$ to the equation $$ Q(\vec{t})=0\,, $$ for which $t_i>0 ~\,\forall i\,$?
In particular I am interested in the case where $M_{i,j} \in \mathbb{Z}$, though this may not be a simplification.
If it is not possible to give complete necessary and sufficient conditions, I would also be interested in sufficient conditions and strong necessary conditions. For example, clearly a necessary condition is that $M$ have elements of opposite sign, and a sufficient condition is that $M$ have diagonal elements of opposite sign. (I have some slightly tighter conditions, but I am wondering if there are significantly stronger conditions.)