3 problems

Weak real zero amalgamation conjecture. There should be a real zero polynomial $Q\in\mathbb{R}[x_i\mid i\in E']$ such that, for $k=1,2$,

Let $\ell=2$, i.e., $x=(x_1,x_2)$, and let $d\in\mathbb N_0$. Suppose $p\in\mathbb R[x,y]$ and $q\in\mathbb R[x,z]$ are real zero polynomials of degree at most $d$ such that…

Schweighofer–Sawall's weak real zero amalgamation conjecture. There should be a real zero polynomial $H\in\mathbb{R}[x_1,x_2,y_1,\ldots,y_m,z_1,\ldots,z_n]$ such that