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(Theorem 5 of ). 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(Theorem 1.4 of ). 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 $$G_T(z,w)=exp(-zw)F(z,-w)$$ and $$G_T(z,-w)=exp(zw)F(z,w)$$. 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 $$G_T(z,-w)=exp(z w)(z+w)$$ and $$G_T(z,w)=exp(-z w)(z-w)$$. 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

• 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  and Theorem 1.4 of , while Theorem 2 from Theorem 1.3 of . The connection between these theorems appeared in the section A.7 of . 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.