Directional hyperbolic families and the obstruction to a uniform leaf

Definitions and mathematical statement

The original source asks only to "extend the theory of interlacing families to several variables" and specifies no unique target theorem. The supplied partial framework uses real homogeneous polynomials as follows. Fix $e\in{\mathbb{R}}^r\setminus\{0\}$. A degree-$m$ homogeneous form $p\in{\mathbb{R}}[x_1,\ldots,x_r]$, with $p(e)>0$, is hyperbolic in direction $e$ if $t\mapsto p(x+te)$ has only real roots for every $x\in{\mathbb{R}}^r$. Write these roots in increasing order as $\lambda_1(p;x),\ldots,\lambda_m(p;x)$, including multiplicity.

For a finite family $p_1,\ldots,p_k$ of the same degree with $p_i(e)>0$, the aid gives the equivalence \[ \text{every }\sum_i c_i p_i\text{ with }c_i\geq0,\ \sum_i c_i>0 \text{ is hyperbolic in }e \] if and only if, for every fixed $x$, the univariate polynomials $p_i(x+te)$ have a common interlacing. Here a common interlacing is a real-rooted degree-$m-1$ polynomial with ordered roots $\mu_1,\ldots,\mu_{m-1}$ such that \[ \lambda_j(p_i;x)\leq\mu_j\leq\lambda_{j+1}(p_i;x) \quad(1\leq j<m,\ 1\leq i\leq k). \] For $m=1$, this condition is vacuous. The interlacing may depend on $x$.

For a parent $p=\sum_i\theta_i p_i$, $\theta_i>0$, $\sum_i\theta_i=1$, this yields, for each fixed $x$, a child satisfying $\lambda_m(p_i;x)\leq\lambda_m(p;x)$. The same fixed-slice conclusion iterates to a leaf in a finite tree with compatible siblings and positive convex parent combinations. It does not assert the quantifier reversal that one child or leaf works for every $x$. Indeed homogeneity gives \[ \lambda_j(p;-x)=-\lambda_{m+1-j}(p;x). \] Thus an upper-largest-root inequality for one child on all slices would also force its smallest root to be no smaller than the parent's on every slice, giving two-sided spectral-interval containment.

In degree one, put $r_i(x)=-p_i(x)/p_i(e)$ and $r(x)=-p(x)/p(e)$, linear root functionals. For a cone $K\subset{\mathbb{R}}^r$, define its dual cone of linear functionals by \[ K^*=\{\ell:\ell(x)\geq0\text{ for all }x\in K\}. \] Then child $i$ has $r_i(x)\leq r(x)$ for all $x\in K$ exactly when $r-r_i\in K^*$. For $K={\mathbb{R}}^r$, this requires $r_i=r$, or equivalently equality of the normalized linear forms.

The aid's nondegenerate quadratic example is \[ p_1(s,z)=s(s-2z),\qquad p_2(s,z)=(s-z)(s+z),\qquad e=(1,0), \] with parent $p=(p_1+p_2)/2=s^2-sz-z^2/2$. Both forms are square-free, positive at $e$, and every nonzero nonnegative combination is hyperbolic in $e$. On $x=(0,1)$, the largest roots of $p_1,p_2,p$ are respectively $2,1,(1+\sqrt3)/2$, so child 1 fails the upper bound. On $x=(0,-1)$, they are $0,1,(-1+\sqrt3)/2$, so child 2 fails. No one child controls all slices. This obstructs one proposed uniform-selection extension, rather than all possible multivariate theories of interlacing.

Short English statement

The source gives a broad request for multivariate interlacing theory. The supplied framework recovers common interlacing and leaf selection on each fixed line, but the chosen leaf can change with the line. A square-free quadratic pair demonstrates why a single uniformly controlling child need not exist; degree one admits an exact dual-cone criterion.

Sources

• Local registry: permanent UnsolvedMath ID $20000548$, exact public code catalogue entry; aim-analysis-notes.json, notes[79].
• AIM workshop Beyond Kadison--Singer: paving and consequences, source item b7: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The exact extracted request is "extend the theory of interlacing families to several variables."
• Supplied research summary, "Directional hyperbolic families and the obstruction to a uniform leaf," attempt 1, describes MSS leaf selection, the compatibility equivalence, dual-cone criterion, and quadratic counterexample. No full MSS citation or other external theorem reference is supplied; original remarks and literature fields are empty.