Let $p$ be a hyperbolic polynomial in the direction $e\in\Bbb R^n$. Then it is also hyperbolic with respect to every direction $v\in\Lambda_{++}:=\{x:p(x-te)=0\implies t>0\}$.
This is from this lecture.
I am reading the proof of Lemma 5 (it's quite short!), and I don't understand this part:
If by increasing $\gamma$ there is ever a root in the upper halfplane, then there must exist a $γ_{\star}$ for which $\beta \mapsto p(\alpha i e+\beta v+\gamma_\star x)$ has a real root $\beta_\star$, and thus $p(\alpha ie+\beta_\star v+\gamma_\star x) = 0$. However, this contradicts hyperbolicity, since $\beta_\star v+\gamma_\star x \in\Bbb R^n$ . Thus, for all $\gamma\ge0$, the roots of $\beta \mapsto p(\alpha i e+\beta v+\gamma x)$ are in the lower halfplane.
I understand the second sentence: from the definition of $p$ being hyperbolic, $t\mapsto p(\beta_\star v+\gamma_\star x-te)$ cannot have $t=-\alpha i\in\Bbb C\backslash\Bbb R$ as a root. But I don't understand the first part.