heorem \[t1\] we are therefore motivated to look closer at the following problem.

\[central\] Let $h$ be a polynomial of degree $d$ which is hyperbolic with respect to ${\mathbf{e}}$, and let $\epsilon >0$ and $m \in {\mathbb{Z}}_+$ be given. Determine the largest possible maximal zero, $\rho=\rho(h,{\mathbf{e}},\epsilon,m)$, of mixed characteristic polynomials: $$\chi[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](t):=h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m](t{\mathbf{e}}+ {\mathbf{1}})$$ subject to the conditions

1.  ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+$,

2.  ${\mathbf{v}}_1 + \cdots + {\mathbf{v}}_m = {\mathbf{e}}$, and

3.  $\tr({\mathbf{v}}_i) \leq \epsilon$ for all $1 \leq i \leq m$.

The following conjecture was made by Marcus *et al.* [@MSS2] in the case when $h = \det$, but we take the liberty to extend the conjecture to any hyperbolic polynomial.

\[maxmax\] The maximal zero in Problem \[central\] is achieved for $${\mathbf{v}}_1=\cdots ={\mathbf{v}}_k= \frac \epsilon d {\mathbf{e}}, {\mathbf{v}}_{k+1}= \left(1-\frac k d \epsilon\right){\mathbf{e}}, {\mathbf{v}}_{k+2}={\mathbf{v}}_{k+3}= \cdots= {\mathbf{v}}_{m}=0,$$ where $k= \lfloor d/\epsilon \rfloor$.

We will prove here that Conjecture \[maxmax\] is equivalent to the following seemingly weaker conjecture.

\[maxmax2\] The maximal zero in Problem \[central\] is achieved for some ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ where ${\mathbf{v}}_i \in \Lambda_{++}\cup \{0\}$ for each $i \in [m]$.

We start by proving that there is a solution to Problem \[central\] for which the ${\mathbf{v}}_i$’s have correct traces, i.e., as those in Conjecture \[maxmax\]. By a “solution" to Problem \[central\] we mean a list of vectors ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$, as in Problem \[central\], which realize the maximal zero. First a useful lemma.

\[nicein\] Suppose ${\mathbf{u}}, {\mathbf{v}}, {\mathbf{w}}\in \Lambda_+$. Then $$(D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}))^2 \geq D_{\mathbf{u}}^2 h({\mathbf{w}}) \cdot D_{\mathbf{v}}^2 h({\math