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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 $f_i$.

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2 Answers 2

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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.

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  • $\begingroup$ The paper you linked to is only about univariate polynomials. (But do you have a reference to a proof of it?) I have asked a related question here on which may be you can shed some light, mathoverflow.net/questions/198354/… $\endgroup$
    – user6818
    Commented Feb 24, 2015 at 23:27
  • $\begingroup$ @Hans Would you know if there is any multivariable analogue of this statement? $\endgroup$
    – guest
    Commented Feb 25, 2015 at 23:40
  • $\begingroup$ @user6818: I think the proof should work with similar techniques as in this paper. at the moment I have no reference. $\endgroup$
    – Hans
    Commented Feb 28, 2015 at 9:25
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    $\begingroup$ @guest: one way to generalize realrootedness to more variables is the concept of hyperbolic polynomials. you can define the interlacing property for them too (for example arxiv.org/abs/1212.6696). and most of what I've written should extend to that case. $\endgroup$
    – Hans
    Commented Feb 28, 2015 at 9:30
  • $\begingroup$ I will look at your related question, but I can't promise anything. $\endgroup$
    – Hans
    Commented Feb 28, 2015 at 9:39
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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)

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