# A counterexample to the Polya-Schur master theorems for half-planes

Given an integer $n\ge 1$ we say that $f\in C[z_1,\ldots,z_n]$ is stable if $f(z_1,\ldots,z_n)\neq 0$ whenever $\text{Im}\ z_i>0$ for all $1\leq i\leq n$.

Stable polynomials with all real coefficients are called real stable. We denote the sets of stable, respectively real stable polyonimals in n variables by $H_n(C)$, respectively $H_n(R)$.

Define the complex Laguerre-Pólya class $\overline{H}_n(C)$ as the class of entire functions in $n$ variables that are limits, uniformly on compact sets, of polynomials in $H_n(C)$. The usual (real) Laguerre-Pólya class $\overline{H}_n(R)$ consists of all functions in $\overline{H}_n(C)$ with real coefficients.

If $T : R[z_1,\ldots, z_n] \rightarrow R[z_1,\ldots,z_n]$, is a linear operator we define its transcendental symbol, $G_T(z,w)$, to be the formal power series in $w_1, \ldots, w_n$ with polynomial coefficients in $R[z_1,\ldots,z_n]$ given by $G_T(z,w) :=\sum_{\alpha \in N^n} (-1)^\alpha T(z^\alpha) \frac {w^\alpha}{\alpha!}.$

A (Weyl algebra) finite order linear partial differential operator with polynomial coefficients is an operator $T: R[z_1,\ldots, z_n] \rightarrow R[z_1,\ldots, z_n]$of the form $T= \sum_{\alpha \leq \beta} Q_\alpha(z) \frac{\partial^\alpha}{\partial z^\alpha}$, where $\beta \in N^n$ and $Q_\alpha\in R[z_1,\ldots, z_n]$, $\alpha\le \beta$.

The followings are two theorems obtained by J.Borcea and P. Brändén.

Theorem 1. Let $T : R[z_1,\ldots, z_n] \rightarrow R[z_1,\ldots, z_n]$ be a linear operator. Then $T$ preserves real stability if and only if either

(a) $T$ has at most $2$-dimensional range and is given by $T(f) = \alpha(f)P + \beta(f)Q,$ where $\alpha, \beta$ are real linear forms on $R[z_1,\ldots, z_n]$ and $P,Q\in H_n(R)$ are such that $P + iQ\in H_n(C)$, or

(b) Either $G_T(z,w)\in \overline{H}_{2n}(R)$ or $G_T(z,-w)\in \overline{H}_{2n}(R)$.

Theorem 2. Let $T: R[z_1,\ldots, z_n] \rightarrow R[z_1,\ldots, z_n]$ be a finite order linear partial differential operator and set $F(z,w) = \sum_{\alpha \leq \beta} Q_\alpha(z)w^\alpha \in R[z_1,\ldots z_n, w_1,\ldots, w_n].$ Then

$T$ preserves real stability if and only if $F(z,-w)$ is real stable.

Question:

A finite order linear partial differential operator is obviously a linear operator and $G_T(z,w)=exp(-z > w)F(z,-w)$, $G_T(z,-w)=exp(z > w)F(z,w)$.

So, if $F(z,w)$ is real stable, does $T$ preserve real stability ? From Theorem 1, it seems that the answer is ture. However, the following counterexample denies it. Let $T: R[z] > \rightarrow R[z]$ be $z+ \frac{d}{d > z}$, then $G_T(z,-w)=exp(z w)F(z,w)$, thus by Theorem 1, $T$ should preserve real stability. But $T(z)=z^2+1$ is not real stable. I am utterly confused by this.

References

1. Julius Borcea and Petter Brändén, Pólya-Schur master theorems for circular domains and their boundaries, Ann. of Math. (2) 170 (2009), no. 1, 465–492. MR 2521123 (2010g:30004), http://dx.doi.org/10.4007/annals.2009.170.465

2. J. Borcea and P. Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. London Math. Soc. 101 (2010), 73-104.

3. Julius Borcea and Petter Brändén, The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability, Invent. Math. 177 (2009), no. 3, 541–569. MR 2534100 (2011g:47069), http://dx.doi.org/10.1007/s00222-009-0189-3

4. Julius Borcea and Petter Brändén, The Lee-Yang and Pólya-Schur programs. II. Theory of stable polynomials and applications, Comm. Pure Appl. Math. 62 (2009), no. 12, 1595–1631. MR 2569072 (2011k:82026), http://dx.doi.org/10.1002/cpa.20295

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Could you please give exact references for Thm 1 and 2? Which paper, which theorem. –  Alexandre Eremenko Mar 27 '13 at 20:12
Of course. Theorem 1 came from Theorem 5 of [1] and Theorem 1.4 of [3], while Theorem 2 from Theorem 1.3 of [2]. The connection between these theorems appeared in the section A.7 of [3]. I expect you could give me a clear answer. –  Pippo Mar 28 '13 at 7:06

Your operator $z+d/dz$ does not preserve real stability. For this operator $F(z,-w)=z-w$ is not real stable. This is consistent with Theorem 2. Concerning Theorem 1, you computed $G(z,w)$ incorrectly. If you compute it correctly, you will see that (b) in Theorem 1 does not hold. So this is not a counterexample.