About preserving real-rootedness of multivariable polynomials Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? 

If necessary assume that,


*

*$\sum_i p_i = 1$

*Each $f_i$ is of bounded degree in each $z_k$. 
Say assume that each $z_k$ occurs with atmost quadratic power in any $f_i$. 
 A: Consider the univariate case:
In general, a linear combination of real-rooted polynomials is not real-rooted: Let $f_1=(z+1)^2$ and $f_2=(z-1)^2$. Every convex combination of $f_1,f_2$ which is not $f_1$ or $f_2$ is not real rooted.
Here are some positive criteria:
Assume that the $f_i$ have a common interlacer and that they have positive leading coeficient, then every convex combination is real-rooted. See http://arxiv.org/abs/1304.4132 for the definitions (text after Lemma 4.2).
If we have just two univariate polynomials $f$ and $g$, then every linear combination is real rooted if and only if $f$ and $g$ are real rooted and interlace each other.
For more than one variable you have to specify what you mean by real rooted: Every nonconstant polynomial in more than one variable has nonreal roots.
A: The complete answer to when and how linear operations on coeffcients of a multivariate polynomial preserve stability is answered in [1] (with more applications of the ideas in [2]), and using those results an answer for the OP should follow.
[1]. Julius Borcea, Petter Brändén. The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability, (2009).
[2]. Julius Borcea, Petter Brändén. The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications, (2008) 
