The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability

Theorem 1.1. Let $\kappa \in \mathbb{N}^n$ and $T : \mathbb{C}_{\kappa}[z_1, \ldots, z_n] \to \mathbb{C}[z_1, \ldots, z_n]$ be a linear operator. Then T preserves stability if and only if either (a) T has range of dimension at most one and is of the form $$T(f) = \alpha(f)P,$$ where $\alpha$ is a linear functional on $\mathbb{C}_{\kappa}[z_1,\ldots,z_n]$ and P is a stable polynomial, or (b) $G_T(z, w) \in \mathcal{H}_{2n}(\mathbb{C})$ . Remark 1.1. Theorem 1.1 may also be stated in terms of other (algebraic) symbols of T that are equivalent to the one defined above, for instance the polynomial $$T[(1-zw)^{\kappa}] = \sum_{\alpha \le \kappa} (-1)^{\alpha} {\kappa \choose \alpha} T(z^{\alpha}) w^{\alpha} = (-1)^{\kappa} w^{\kappa} G_T(z, -w^{-1}).$$ Note that $T[(1-zw)^{\kappa}] \in \mathcal{H}_{2n}(\mathbb{C}) \Leftrightarrow G_T(z,w) \in \mathcal{H}_{2n}(\mathbb{C})$ , see, e.g., Lemma 1.7 (3).

Theorem 1.2. Let $\kappa \in \mathbb{N}^n$ and $T : \mathbb{R}_{\kappa}[z_1, \ldots, z_n] \to \mathbb{R}[z_1, \ldots, z_n]$ be a linear operator. Then T preserves real stability if and only if either (a) T has range of dimension no greater than two and is of the form $$T(f) = \alpha(f)P + \beta(f)Q,$$ where $\alpha, \beta : \mathbb{R}_{\kappa}[z_1, \dots, z_n] \to \mathbb{R}$ are linear functionals and P, Q are real stable polynomials such that $P \ll Q$ , or - (b) $G_T(z, w) \in \mathcal{H}_{2n}(\mathbb{R})$ , or - (c) $G_T(z, -w) \in \mathcal{H}_{2n}(\mathbb{R})$ . If $T: \mathbb{K}[z_1,\ldots,z_n] \to \mathbb{K}[z_1,\ldots,z_n]$ , where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ , is a linear operator we define its $transcendental\ symbol,\ \overline{G}_T(z,w)$ , to be the formal power series in $w_1,\ldots,w_n$ with polynomial coefficients in $\mathbb{K}[z_1,\ldots,z_n]$ given by $$\overline{G}_T(z,w) = \sum_{\alpha \in \mathbb{N}^n} (-1)^{\alpha} T(z^{\alpha}) \frac{w^{\alpha}}{\alpha!},$$ where $\alpha! = \alpha_1! \cdots \alpha_n!$ . By abuse of notation we write $\overline{G}_T(z, w) = T[e^{-z \cdot w}]$ , where $z \cdot w = z_1 w_1 + \ldots + z_n w_n$ . Let us define the complex Laguerre-Pólya class $\overline{\mathcal{H}}_n(\mathbb{C})$ as the class of entire functions in n variables that are limits, uniformly on compact sets, of polynomials in $\mathcal{H}_n(\mathbb{C})$ , see, e.g., [32, Chap. IX]. The usual Laguerre-Pólya class $\overline{\mathcal{H}}_n(\mathbb{R})$ consists of all functions in $\overline{\mathcal{H}}_n(\mathbb{C})$ with real coefficients. Our next theorem provides a transcendental characterization of stability preservers on infinite-dimensional complex polynomial spaces, thus solving Problem 2 for $\mathbb{K}=\mathbb{C}$ and $\Omega=H$ .

Theorem 1.3. Let $T: \mathbb{C}[z_1,\ldots,z_n] \to \mathbb{C}[z_1,\ldots,z_n]$ be a linear operator. Then T preserves stability if and only if either (a) T has range of dimension at most one and is of the form $$T(f) = \alpha(f)P$$ , where $\alpha$ is a linear functional on $\mathbb{C}[z_1,\ldots,z_n]$ and P is a stable polynomial, (b) $$\frac{or}{\overline{G}_T(z,w)} \in \overline{\mathcal{H}}_{2n}(\mathbb{C}).$$ Remark 1.2. From Theorem 1.3 one can easily deduce a characterization of linear operators preserving $\Omega$ -stability for any open half-plane $\Omega$ . For instance, the analog of Theorem 1.3 (b) for the open right half-plane $\mathbb{H}_{\frac{\pi}{2}}$ is that the transcendental symbol of T with respect to $\mathbb{H}_{\frac{\pi}{2}}$ , i.e., the formal power series $$T[e^{z \cdot w}] := \sum_{\alpha \in \mathbb{N}^n} T(z^{\alpha}) \frac{w^{\alpha}}{\alpha!},$$ defines an entire function which is the limit, uniformly on compact sets, of weakly Hurwitz stable polynomials. The analog of Theorem 1.3 for real stability presevers – which solves Problem 2 for $\mathbb{K} = \mathbb{R}$ and $\Omega = H$ – is as follows.

Theorem 1.4. Let $T: \mathbb{R}[z_1, \ldots, z_n] \to \mathbb{R}[z_1, \ldots, z_n]$ be a linear operator. Then Tpreserves real stability if and only if either (a) T has range of dimension no greater than two and is of the form $$T(f) = \alpha(f)P + \beta(f)Q,$$ where $\alpha, \beta : \mathbb{R}[z_1, \dots, z_n] \to \mathbb{R}$ are linear functionals and P, Q are real stable polynomials such that $P \ll Q$ , or - (b) $G_T(z, w) \in \mathcal{H}_{2n}(\mathbb{R})$ , or - (c) $\overline{G}_T(z,-w) \in \overline{\mathcal{H}}_{2n}(\mathbb{R})$ . - 1.2. Fundamental Properties of Stable Polynomials. In this section we first review some basic facts and then prove several results on (real) stable polynomials needed later on. We start with an immediate consequence of the definitions (cf. [9, Lemma 2.1]).

Lemma 1.5. Let $f \in \mathbb{K}[z_1, \ldots, z_n]$ , where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ . Then $f \in \mathcal{H}_n(\mathbb{K})$ if and only if $f(\lambda t + \alpha) \in \mathcal{H}_1(\mathbb{K})$ for all $\lambda \in \mathbb{R}^n_+$ and $\alpha \in \mathbb{R}^n$ . The following theorem is a multivariate version of Hurwitz' theorem on the "continuity of zeros", see, e.g., [13, Footnote 3, p. 96] for a

Theorem 1.6. (Hurwitz' theorem). Let D be a domain (open connected set) in $\mathbb{C}^n$ and suppose $\{f_k\}_{k=1}^{\infty}$ is a sequence of non-vanishing analytic functions on D that converge to f uniformly on compact subsets of D. Then f is either non-vanishing on D or else identically zero. We next list some of the closure properties for (real) stable polynomials. The first property in Lemma 1.7 below is deduced by applying Theorem 1.6 with D = $\{z \in \mathbb{C} : \text{Im}(z) > 0\}^n \text{ and } f_k(z_1, \dots, z_n) = f(z_1, \dots, z_{i-1}, \mu + z_i/k, z_{i+1}, \dots, z_n),$ $k \in \mathbb{N}$ , while the remaining properties are easy consequences of the definitions and Lemma 1.5.

Lemma 1.7. Let $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ and $f \in \mathcal{H}_n(\mathbb{K})$ be of degree $d_j$ in $z_j$ , $1 \leq j \leq n$ . Then for any $1 \le i \le n$ one has: - (1) $f(z_1, \ldots, z_{i-1}, \mu, z_{i+1}, \ldots, z_n) \in \mathcal{H}_{n-1}(\mathbb{K}) \cup \{0\} \text{ for } \mu \in \mathbb{R};$ - (2) $f(z_1, \dots, z_{i-1}, \lambda z_i, z_{i+1}, \dots, z_n) \in \mathcal{H}_n(\mathbb{K})$ for $\lambda > 0$ ; (3) $z_i^{d_i} f(z_1, \dots, z_{i-1}, -z_i^{-1}, z_{i+1}, \dots, z_n) \in \mathcal{H}_n(\mathbb{K})$ ; (4) $f(z_1, \dots, z_{i-1}, z_j, z_{i+1}, \dots, z_n) \in \mathcal{H}_{n-1}(\mathbb{K})$ for $1 \le j \ne i \le n$ . We will now establish a series of results involving the notion of proper position introduced in Definition 1.1, compare with [9, §2], [13, Theorem 3.2]. In particular, the next lemma provides a multivariate analog of the classical Hermite-Biehler theorem showing that proper position is a natural higher dimensional counterpart of interlacing.

Lemma 1.8. Let $f, g \in \mathbb{R}[z_1, \dots, z_n] \setminus \{0\}$ , set h = f + ig and suppose that f and g are not constant multiples of each other. The following are equivalent: - (1) $h \in \mathcal{H}_n(\mathbb{C})$ , that is, $g \ll f$ ; - (2) $|h(z)| > |h(\bar{z})|$ for all $z = (z_1, \ldots, z_n) \in \mathbb{C}^n$ with $\text{Im}(z_j) > 0, 1 \le j \le n$ , where $\bar{z} = (\bar{z}_1, \ldots, \bar{z}_n)$ ; - (3) $f + z_{n+1}g \in \mathcal{H}_{n+1}(\mathbb{R});$ - (4) $f, g \in \mathcal{H}_n(\mathbb{R})$ and $$\operatorname{Im}\left(\frac{f(z)}{g(z)}\right) \ge 0$$ whenever $z = (z_1, ..., z_n) \in \mathbb{C}^n$ with $\text{Im}(z_j) > 0$ , $1 \le j \le n$ ; (5) $g(\lambda t + \alpha) \ll f(\lambda t + \alpha)$ for all $\lambda \in \mathbb{R}^n_+$ and $\alpha \in \mathbb{R}^n$ .

Theorem 1.9. Let $f, g \in \mathbb{R}[z_1, \ldots, z_n]$ . All non-zero polynomials in the space $$\{\alpha f + \beta g : \alpha, \beta \in \mathbb{R}\}\$$ are stable if and only if either $f = g \equiv 0$ , $f \ll g$ or $g \ll f$ . Moreover, if $g \ll f$ then $W_j[g,f](x) \leq 0$ for all $x \in \mathbb{R}^n$ and $1 \leq j \leq n$ , and if $f \ll g$ then $W_j[g,f](x) \geq 0$ for all $x \in \mathbb{R}^n$ and $1 \leq j \leq n$ .

Corollary 1.10. If $f, g \in \mathbb{R}[z_1, \dots, z_n] \setminus \{0\}$ are real stable polynomials such that $f \ll g$ and $g \ll f$ then $f = \alpha g$ for some $\alpha \in \mathbb{R}$ .

Lemma 2.1. (Lieb-Sokal). Let $P(z) + wQ(z) \in \mathbb{C}[z_1, \ldots, z_n, w]$ be stable. If the degree in the variable $z_i$ is at most one then the polynomial $$P(z) - \frac{\partial Q(z)}{\partial z_i}$$ is either identically zero or stable.

Lemma 2.2. Let $T: \mathbb{C}_{(1^n)}[z_1,\ldots,z_n] \to \mathbb{C}[z_1,\ldots,z_n]$ be a linear operator such that $$G_T(z, w) := T\left[(z+w)^{[n]}\right] = \sum_{S\subseteq[n]} T\left[z^S\right] w^{[n]\setminus S}$$ is stable, where $z = (z_1, \ldots, z_m)$ and $w = (w_1, \ldots, w_n)$ . Then T preserves stability.

Theorem 2.3. (Grace-Walsh-Szegö). Let f be a symmetric multi-affine polynomial in n complex variables, let C be an open or closed circular domain, and let $\xi_1, \ldots, \xi_n$ be points in C. Suppose further that either the total degree of f equals n or that C is convex (or both). Then there exists at least one point $\xi \in C$ such that $$f(\xi_1, \dots, \xi_n) = f(\xi, \dots, \xi). \tag{2.3}$$ We give a new and self-contained proof of Theorem 2.3 in [6, §2].

Proposition 2.4. Let $f \in \mathbb{C}_{\kappa}[z_1, \ldots, z_n]$ . Then f is stable if and only if $\Pi_{\kappa}^{\uparrow}(f)$ is stable.

Lemma 2.5. Let $T: \mathbb{C}_{\kappa}[z_1,\ldots,z_n] \to \mathbb{C}_{\gamma}[z_1,\ldots,z_n]$ be a linear operator. Then the symbol of the polarization of T is the polarization of the symbol of T, that is, $$G_{\Pi(T)} = \Pi^{\uparrow}_{\gamma \oplus \kappa}(G_T),$$ where $\Pi_{\gamma \oplus \kappa}^{\uparrow} : \mathbb{C}_{\gamma \oplus \kappa}[z_1, \dots, z_n, w_1, \dots, w_n] \to \mathbb{C}_{MA}^{\gamma \oplus \kappa}$

Lemma 3.1. Let $f \in \mathbb{C}_{\kappa}[z_1, \ldots, z_n]$ , where $\kappa = (\kappa_1, \ldots, \kappa_n) \in \mathbb{N}^n$ , and $W = (W_1, \ldots, W_n) \in \{\zeta \in \mathbb{C} : \operatorname{Im}(\zeta) > 0\}^n$ . Then for all $\epsilon > 0$ sufficiently small one has $(z+W)^{\kappa} + \epsilon f(z) \in \mathcal{H}_n(\mathbb{C})$ , where $(z+W)^{\kappa} = (z_1+W_1)^{\kappa_1} \cdots (z_n+W_n)^{\kappa_n}$ .

Lemma 3.2. Let $V \subseteq \mathbb{K}[z_1, \ldots, z_n]$ be a $\mathbb{K}$ -linear space, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ . - (i) If $\mathbb{K} = \mathbb{R}$ and every non-zero element of V is real stable then dim $V \leq 2$ . - (ii) If $\mathbb{K} = \mathbb{C}$ and every non-zero element of V is stable then dim $V \leq 1$ . Proof. We first deal with the real case. Suppose that there are three linearly independent polynomials $f_1, f_2$ and $f_3$ in V. By the multivariate Hermite-Kakeya-Obreschkoff theorem (Theorem 1.9) and the assumption on V these polynomials are mutually in proper position. Without loss of generality we may assume that $f_1 \ll f_2$ and $f_1 \gg f_3$ . Now consider the line segment in V given by $\ell_\theta = \theta f_3 + (1-\theta)f_2$ , $0 \le \theta \le 1$ , and note in particular that for any such $\theta$ one has either $f_1 \ll \ell_\theta$ or $f_1 \gg \ell_\theta$ . Set $\eta = \sup\{\theta \in [0,1]: f_1 \ll \ell_\theta\}$ . Since $f_1 \ll \ell_0$ and $f_1 \gg \ell_1$ it follows from Hurwitz' theorem (Theorem 1.6) that $f_1 \ll \ell_\eta$ and $f_1 \gg \ell_\eta$ . This means that $f_1$ and $\ell_\eta$ are constant multiples of each other (cf. Corollary 1.10), contrary to the assumption that $f_1, f_2$ and $f_3$ are linearly independent. For the complex case let $$V_R = \{p : p + iq \in V \text{ with } p, q \in \mathbb{R}[z_1, \dots, z_n]\}$$ be the "real component" of V. By Theorem 1.9 all non-zero polynomials in $V_R$ are real stable hence $\dim_{\mathbb{R}} V_R \leq 2$ by part (i) proved above. Note that V is the complex span of $V_R$ . If $\dim_{\mathbb{R}} V_R \leq 1$ we are done so we may assume that $\{p,q\}$ is a basis of $V_R$ with $f := p + iq \in V$ which by assumption is a (not identically zero) stable polynomial. By Theorem 1.9 again we have $W_j[p,q](x) \geq 0$ , $1 \leq j \leq n$ , $x \in \mathbb{R}^n$ , and since p and q are linearly independent the $j_0$ -th Wronskian $W_{j_0}[p,q]$ is not identically zero for some index $j_0$ . Now if $g \in V \setminus \{0,f\}$ we may write $$q = ap + bq + i(cp + dq)$$ for some $a, b, c, d \in \mathbb{R}$ . We have to show that g is a (complex) constant multiple of f. Since $g \in V$ is not identically zero it is a stable polynomial, so that in particular $$W_{j_0}[ap + bq, cp + dq] = (ad - bc)W_{j_0}[p, q] \ge 0$$ (cf. Theorem 1.9). Since $W_{j_0}[p,q] \ge 0$ and $W_{j_0}[p,q] \not\equiv 0$ it follows that $ad - bc \ge 0$ . Let $u,v \in \mathbb{R}$ and note that by linearity we have $$g + (u + iv)f = (a + u)p + (b - v)q + i((c + v)p + (d + u)q) \in V.$$ Arguing as above we deduce that $$H(u,v) := (a+u)(d+u) - (b-v)(c+v) > 0$$ for all $u, v \in \mathbb{R}$ . But $$4H(u,v) = (2u+a+d)^2 + (2v+c-b)^2 - (a-d)^2 - (b+c)^2,$$ so $H(u,v) \geq 0$ for all $u,v \in \mathbb{R}$ if and only if a=d and b=-c. This gives $$g = ap - cq + i(cp + aq) = (a + ic)(p + iq) = (a + ic)f,$$ as was to be shown. We can now show that the symbol of any stability preserver whose range is not one-dimensional must necessarily be stable. Proof of Necessity in Theorem 1.1. Suppose that $T: \mathbb{C}_{\kappa}[z_1,\ldots,z_n] \to \mathbb{C}[z_1,\ldots,z_n]$ preserves stability. Given $W \in \{\zeta \in \mathbb{C} : \operatorname{Im}(\zeta) > 0\}^n$ we have that $T[(z+W)^{\kappa}] \in \mathcal{H}_n(\mathbb{C}) \cup \{0\}$ . Assume first that there exists $W \in \{\zeta \in \mathbb{C} : \operatorname{Im}(\zeta) > 0\}^n$ for which $T[(z+W)^{\kappa}] \equiv 0$ and let $f \in \mathbb{C}_{\kappa}[z_1,\ldots,z_n]$ . Then by Lemma 3.1 there is an $\epsilon > 0$ such that $(z+W)^{\kappa} + \epsilon f(z) \in \mathcal{H}_n(\mathbb{C})$ . It follows that $$\epsilon T[f(z)] = T[(z+W)^{\kappa} + \epsilon f(z)] \in \mathcal{H}_n(\mathbb{C}) \cup \{0\}.$$ Hence the image of T is a linear space whose non-zero elements are all stable polynomials. By Lemma 3.2 (ii) we have that T is given by $T(f) = \alpha(f)P$ , where P is a stable polynomial and $\alpha$ is a linear functional. On the other hand, if $T[(z+W)^{\kappa}] \not\equiv 0$ for $W \in \{\zeta \in \mathbb{C} : \operatorname{Im}(\zeta) > 0\}^n$ then $T[(z+W)^{\kappa}] \in \mathcal{H}_n(\mathbb{C})$ for all such W, so that $G_T(z,w) = T[(z+w)^{\kappa}] \in \mathcal{H}_{2n}(\mathbb{C})$ . $\square$

Proposition 4.1. For any $n \in \mathbb{N}$ the following holds: $$\mathcal{H}_n(\mathbb{C}) \cap \mathcal{H}_n^-(\mathbb{C}) = \mathbb{C}\mathcal{H}_n(\mathbb{R}) := \{cf : c \in \mathbb{C}, f \in \mathcal{H}_n(\mathbb{R})\}.$$

Theorem 5.1. Let $F(z,w) = \sum_{\alpha \in \mathbb{N}^n} P_{\alpha}(z) w^{\alpha}$ , where $z = (z_1, \ldots, z_n)$ and $w = (w_1, \ldots, w_n)$ , be a formal power series in w with coefficients in $\mathbb{K}[z_1, \ldots, z_n]$ . Then $F(z,w) \in \overline{\mathcal{H}}_{2n}(\mathbb{K})$ if and only if $$\sum_{\alpha \leq \beta} (\beta)_{\alpha} P_{\alpha}(z) w^{\alpha} \in \mathcal{H}_{2n}(\mathbb{K}) \cup \{0\}$$ for all $\beta \in \mathbb{N}^n$ . The proof of Theorem 5.1 requires several new ingredients and additional results that we proceed to describe. Since the arguments are the same for $\mathbb{K} = \mathbb{R}$ and $\mathbb{K} = \mathbb{C}$ , we will only focus on the latter case. 5.1. Generalized Jensen Multipliers. For $\alpha, \beta \in \mathbb{N}^n$ let $J(\alpha, \beta) = (\beta)_{\alpha}\beta^{-\alpha}$ (we use the convention that $k^{\pm k} = 1$ for k = 0). For n = 1 these are the so-called Jensen multipliers that are known to preserve (univariate) stability [14, 32, 43]. For each fixed $\beta \in \mathbb{N}^n$ the sequences $\{J(\alpha, \beta)\}_{\alpha \leq \beta}$ and $\{(\beta)_{\alpha}\}_{\alpha \leq \beta}$ may be called multivariate (or generalized) Jensen multipliers in view of the following lemma.

Lemma 5.2. Let $\beta \in \mathbb{N}^n$ . The linear operators on $\mathbb{C}[z_1, \ldots, z_n]$ defined by $$z^{\alpha} \mapsto J(\alpha, \beta) z^{\alpha}, \quad \alpha \in \mathbb{N}^n,$$ $z^{\alpha} \mapsto (\beta)_{\alpha} z^{\alpha}, \quad \alpha \in \mathbb{N}^n,$ preserve stability.

Lemma 5.3. (Szász). Suppose that $f(z) = 1 + \sum_{i=1}^k a_i z^i = \prod_{j=1}^k (1 + \xi_j z)$ is stable. Then $$\sum_{j=1}^{k} |\xi_j|^2 \le 3|a_1|^2 + 2|a_2|.$$

Lemma 5.4. Suppose that the polynomial $$f(z) = 1 + \sum_{|\beta| > 0} a(\beta)z^{\beta} \in \mathbb{C}[z_1, \dots, z_n]$$ is stable and let $$A = \left[ 3 \left( \sum_{i=1}^{n} |a(e_i)| \right)^2 + 2 \sum_{i,j=1}^{n} |a(e_i + e_j)| \right]^{1/2}.$$ Then $$|a(\beta)| \le |\beta|^{-|\beta|/2} \frac{\beta^{\beta}}{\beta!} A^{|\beta|}, \quad \beta = (\beta_1, \dots, \beta_n) \in \mathbb{N}^n,$$ where $|\beta| = \sum_{i=1}^{n} \beta_i$ .

Theorem 5.5. Suppose that the polynomial $$f(z) = 1 + \sum_{|\beta| > 0} a(\beta)z^{\beta} \in \mathbb{C}[z_1, \dots, z_n]$$ is stable and let $$B = 2^{n-1} \frac{\sqrt{2e^2 - e}}{e - 1} = 2^{n-1} \cdot 2.0210 \dots,$$ $$C = 6e^2 \left( \sum_{i=1}^n |a(e_i)| \right)^2 + 4e^2 \sum_{i,j=1}^n |a(e_i + e_j)|.$$ Then $$\max \{ |f(z)| : |z_i| \le r, 1 \le i \le n \} \le Be^{Cr^2}, \quad r \ge 0.$$

Theorem 5.6. Let $\mathcal{M} \subset \mathbb{N}^n$ be a finite non-empty set and $f(z) = \sum_{\alpha} a(\alpha) z^{\alpha} \in \mathbb{C}[z_1, \ldots, z_n]$ be a stable polynomial with $\mathcal{M}(f) = \mathcal{M}$ . Then there are constants B and C depending only on the coefficients $a(\alpha)$ with $\alpha \in \mathcal{M}_2$ such that $$\max\left\{|f(z)|:|z_i|\leq r, 1\leq i\leq n\right\}\leq Be^{Cr^2}$$ for all $r \geq 0$ . Moreover, B and C can be chosen so that they depend continuously on the aforementioned set of coefficients.

Lemma 6.1. Let $\{C_i\}_{i=1}^n$ be a family of circular domains, $f \in \mathbb{C}[z_1,\ldots,z_n]$ be of degree $\kappa \in \mathbb{N}^n$ , and $J \subseteq [n]$ a (possibly empty) set such that $C_j$ is the exterior of a disk whenever $j \in J$ . Denote by g the polynomial in the variables $z_j$ , $j \in J$ , obtained by setting $z_i = c_i \in C_i$ arbitrarily for $i \notin J$ . If f is $C_1 \times \cdots \times C_n$ -stable then supp(g) has a unique maximal element $\gamma$ with respect to the standard partial order on $\mathbb{N}^J$ . Moreover, $\gamma$ is the same for all choices of $c_i \in C_i$ , $i \notin J$ .

Lemma 6.2. Suppose that $C_1, \ldots, C_n, D_1, \ldots, D_n$ are open circular domains and $\kappa = (\kappa_1, \ldots, \kappa_n) \in \mathbb{N}^n$ . Then there are Möbius transformations $$\zeta \mapsto \phi_i(\zeta) = \frac{a_i \zeta + b_i}{c_i \zeta + d_i}, \quad i \in [n],$$ (6.2) as in (6.1) such that the (invertible) linear transformation $\Phi_{\kappa}: \mathbb{C}_{\kappa}[z_1,\ldots,z_n] \to \mathbb{C}_{\kappa}[z_1,\ldots,z_n]$ defined by $$\Phi_{\kappa}(f)(z_1,\ldots,z_n) = (c_1 z_1 + d_1)^{\kappa_1} \cdots (c_n z_n + d_n)^{\kappa_n} f(\phi_1(z_1),\ldots,\phi_n(z_n))$$ (6.3) restricts to a bijection between $\mathcal{N}_{\kappa}(C_1,\ldots,C_n)$ and $\mathcal{N}_{\kappa}(D_1,\ldots,D_n)$ .

Theorem 6.3. Let $\kappa \in \mathbb{N}^n$ and $T : \mathbb{C}_{\kappa}[z_1, \ldots, z_n] \to \mathbb{C}[z_1, \ldots, z_n]$ be a linear operator. Let further $C_i$ , $i \in [n]$ , be open circular domains given by $C_i = \phi_i^{-1}(H)$ , $i \in [n]$ , where $\phi_i$ , $i \in [n]$ , are Möbius transformations as in (6.2) such that the corresponding linear transformation $\Phi_{\kappa}$ defined in (6.3) restricts to a bijection between $\mathcal{N}_{\kappa}(H, \ldots, H)$ and $\mathcal{N}_{\kappa}(C_1, \ldots, C_n)$ (cf. Lemma 6.2). Then $$T: \mathcal{N}_{\kappa}(C_1, \dots, C_n) \to \mathcal{N}(C_1, \dots, C_n) \cup \{0\}$$ if and only if either (a) T has range of dimension at most one and is of the form $$T(f) = \alpha(f)P$$ . where $\alpha$ is a linear functional on $\mathbb{C}_{\kappa}[z_1,\ldots,z_n]$ and P is a $C_1\times\cdots\times C_n$ -stable polynomial, or (b) the polynomial in 2n variables $z_1, \ldots, z_n, w_1, \ldots, w_n$ given by $$T\left[\prod_{i=1}^{n} ((a_{i}z_{i}+b_{i})(c_{i}w_{i}+d_{i})+(a_{i}w_{i}+b_{i})(c_{i}z_{i}+d_{i}))^{\kappa_{i}}\right]$$ is $$C_1 \times \cdots \times C_n \times C_1 \times \cdots \times C_n$$ -stable. Remark 6.1. By analogy with the definitions made in §1.1 we may call the polynomial in Theorem 6.3 (b) the algebraic symbol of T with respect to $C_1 \times \cdots \times C_n$ . Note that in the case of the open unit disk $C_i = \mathbb{D}$ , $i \in [n]$ , or the exterior of the closed unit disk $C_i = \mathbb{C} \setminus \overline{\mathbb{D}}$ , $i \in [n]$ , this symbol becomes a constant multiple of $T[(1+zw)^{\kappa}]$ while in the case of an open half-plane bordering on the origin it is just a constant multiple of $G_T(z,w) = T[(z+w)^{\kappa}]$ . For the open right half-plane $\mathbb{H}_{\frac{\pi}{2}}$ it is often more convenient (but equivalent) to choose the symbol $T[(1+zw)^{\kappa}]$ . Proof of Theorem 6.3. Let $T: \mathbb{C}_{\kappa}[z_1,\ldots,z_n] \to \mathbb{C}[z_1,\ldots,z_n]$ be a linear operator and $\gamma = (\gamma_1,\ldots,\gamma_n) \in \mathbb{N}^n$ be such that $T(\mathbb{C}_{\kappa}[z_1,\ldots,z_n]) \subseteq \mathbb{C}_{\gamma}[z_1,\ldots,z_n]$ . The theorem is trivial if the range of T has dimension at most one so we may assume that this is not the case. With $\phi_i, i \in [n]$ , and $\Phi_{\kappa}$ as in the statement of the theorem let $\Phi_{\gamma}: \mathbb{C}_{\gamma}[z_1,\ldots,z_n] \to \mathbb{C}_{\gamma}[z_1,\ldots,z_n]$ be the linear transformation defined as in (6.3) only with $\gamma = (\gamma_1,\ldots,\gamma_n)$ instead of $\kappa = (\kappa_1,\ldots,\kappa_n)$ . Set $T':=\Phi_{\gamma}^{-1}T\Phi_{\kappa}$ . By Theorem 1.1 and Lemma 6.2 we have the following chain of equivalences: $$T(\mathcal{N}_{\kappa}(C_1,\ldots,C_n)) \subset \mathcal{N}(C_1,\ldots,C_n) \cup \{0\} \iff$$ $$T' \text{ preserves } (H\text{-)stability} \iff$$ $$G(z,w) := T'[(z+w)^{\kappa}] \text{ is } (H\text{-)stable} \iff$$ $$\Phi_{\gamma \oplus \kappa}(G) \text{ is } C_1 \times \cdots \times C_n \times C_1 \times \cdots \times C_n\text{-stable,}$$ where $\Phi_{\gamma \oplus \kappa} : \mathbb{C}_{\gamma \oplus \kappa}[z_1, \dots, z_n, w_1, \dots, w_n] \to \mathbb{C}_{\gamma \oplus \kappa}[z_1, \dots, z_n, w_1, \dots, w_n]$ is given by $$\Phi_{\gamma \oplus \kappa}(f)(z_1, \dots, z_n, w_1, \dots, w_n)$$ $$= \prod_{i=1}^n (c_i z_i + d_i)^{\gamma_i} \prod_{i=1}^n (c_i w_i + d_i)^{\kappa_i} f(\phi_1(z_1), \dots, \phi_n(z_n), \phi_1(w_1), \dots, \phi_n(w_n)),$$ see (6.3). Now $\Phi_{\gamma \oplus \kappa}(G)$ is precisely the polynomial in Theorem 6.3 (b). Indeed, letting subscript z and subscript w indicate that the corresponding operator acts on the z and w variables, respectively, we have $$\Phi_{\gamma \oplus \kappa}(G)(z, w) = \left(\Phi_{\kappa, w} \circ \Phi_{\gamma, z} \circ \Phi_{\gamma, z}^{-1} \circ T_{z} \circ \Phi_{\kappa, z}\right) \left[ (z + w)^{\kappa} \right] = T \left[ \Phi_{\kappa \oplus \kappa} \left( (z + w)^{\kappa} \right) \right],$$ which completes the proof of the theorem. Note that if all $C_i$ are open convex circular domains then Theorem 6.3 answers Problem 1 (for such domains) precisely as stated in the introduction, and when non-convex circular domains are involved it answers a more precise version of the problem. 6.2. Linear Preservers of Lee-Yang Polynomials. Given an open circular domain $C \subset \mathbb{C}$ let $C^r$ be the interior of the complement of C, that is, $C^r = \mathbb{C} \setminus \overline{C}$ .

Theorem 6.4. Let $\kappa \in \mathbb{N}^n$ , $T : \mathbb{C}_{\kappa}[z_1, \ldots, z_n] \to \mathbb{C}[z_1, \ldots, z_n]$ be a non-degenerate linear operator and $C_i$ , $i \in [n]$ , be open circular domains given by $C_i = \phi_i^{-1}(H)$ , $i \in [n]$ , where $\phi_i$ , $i \in [n]$ , are Möbius transformations as in Theorem 6.3. Then T preserves the $\kappa$ -Lee-Yang property with respect to $\{C_i\}_{i=1}^n$ if and only if either (a) the polynomial in 2n variables $z_1, \ldots, z_n, w_1, \ldots, w_n$ given by $$T\left[\prod_{i=1}^{n} ((a_{i}z_{i}+b_{i})(c_{i}w_{i}+d_{i})+(a_{i}w_{i}+b_{i})(c_{i}z_{i}+d_{i}))^{\kappa_{i}}\right]$$ is $C_1 \times \cdots \times C_n \times C_1 \times \cdots \times C_n$ -stable and $C_1^r \times \cdots \times C_n^r \times C_1^r \times \cdots \times C_n^r$ -stable, or (b) the polynomial in 2n variables $z_1, \ldots, z_n, w_1, \ldots, w_n$ given by $$T\left[\prod_{i=1}^{n} ((a_{i}z_{i}+b_{i})(c_{i}w_{i}+d_{i})-(a_{i}w_{i}+b_{i})(c_{i}z_{i}+d_{i}))^{\kappa_{i}}\right]$$ is $$C_1 \times \cdots \times C_n \times C_1 \times \cdots \times C_n$$ -stable and $C_1^r \times \cdots \times C_n^r \times C_1^r \times \cdots \times C_n^r$ -stable.

Theorem 7.1. Let $\kappa \in \mathbb{N}^n$ and $T : \mathbb{C}_{\kappa}[z_1, \ldots, z_n] \to \mathbb{C}[z_1, \ldots, z_n]$ be a linear operator. If $G_T(z, w)$ is a strictly stable polynomial then T preserves strict stability. One can also check that the sufficiency part of Theorem 6.3 carries over to closed circular domains and thus yields a (sufficient) condition for linear operators $T: \mathbb{C}_{\kappa}[z_1,\ldots,z_n] \to \mathbb{C}[z_1,\ldots,z_n]$ to preserve $\overline{C}_1 \times \cdots \times \overline{C}_n$ -stability, where $C_i$ , $i \in [n]$ , are arbitrary (open) circular domains. For simplicity we only state here a special case that extends Theorems 7.1.

Theorem 7.2. Let $\kappa \in \mathbb{N}^n$ , $T : \mathbb{C}_{\kappa}[z_1, \ldots, z_n] \to \mathbb{C}[z_1, \ldots, z_n]$ a linear operator and $\overline{C}$ a convex closed circular domain given by $\overline{C} = \phi^{-1}(\overline{H})$ , where $\phi$ is a Möbius transformation as in (6.1). If $$T\left[\left((az+b)(cw+d)+(aw+b)(cz+d)\right)^{\kappa}\right]$$ is $\overline{C}$ -stable then T preserves $\overline{C}$ -stability. As noted in Remark 6.1, for the the closed unit disk $\overline{C} = \overline{\mathbb{D}}$ the symbol used in Theorem 7.2 is just a constant multiple of $T[(1+zw)^{\kappa}]$ while for a closed half-plane bordering on the origin it is a constant multiple of $G_T(z,w) = T[(z+w)^{\kappa}]$ . Note though that the conditions in Theorems 7.1, 7.2 are not necessary. For instance, the identity operator obviously preserves strict stability but its (algebraic) symbol $(z+w)^{\kappa}$ is not strictly stable, cf. (i) in §8.

Definition 1.1. Two polynomials $f, g \in \mathbb{R}[z_1, \dots, z_n]$ are in proper position, denoted $f \ll g$ , if $g + if \in \mathcal{H}_n(\mathbb{C})$ . By the Hermite-Biehler theorem (see, e.g., [43, Theorem 6.3.4]) in the univariate case the relation $f \ll g$ is equivalent to saying that f and g are real-rooted (or identically zero), their zeros interlace and their $Wronskian\ W[f,g] := f'g - fg'$ is non-positive on the whole of $\mathbb{R}$ . We study further properties of this relation in §1.2. The algebraic characterization of real stability preservers on finite-dimensional polynomial spaces – which solves Problem 1 for $\mathbb{K} = \mathbb{R}$ and $\Omega = H$ – is as follows.

Definition 1.2. For $g, f \in \mathbb{R}[z_1, \dots, z_n]$ and $1 \leq j \leq n$ define the j-th Wronskian of g, f as $$W_j[g, f] = \frac{\partial g}{\partial z_j} \cdot f - g \cdot \frac{\partial f}{\partial z_j}.$$ The following multivariate analog of the Hermite-Kakeya-Obreschkoff theorem (see, e.g., [43, Theorem 6.3.8] for the classical univariate version) was first proved in [9]. Below we give an alternative shorter

Definition 6.1. Let $\kappa = (\kappa_1, \dots, \kappa_n) \in \mathbb{N}^n$ and $\{C_i\}_{i=1}^n$ be a family of open circular domains. The set of $\kappa$ -Lee-Yang polynomials with respect to $\{C_i\}_{i=1}^n$ is $$\mathcal{LY}_{\kappa}(C_1,\ldots,C_n) = \mathcal{N}_{\kappa}(C_1,\ldots,C_n) \cap \mathcal{N}_{\kappa}(C_1^r,\ldots,C_n^r).$$ A Lee-Yang polynomial with respect to $\{C_i\}_{i=1}^n$ is an element of the set $$\mathcal{L}\mathcal{Y}(C_1,\ldots,C_n) := \bigcup_{\kappa \in \mathbb{N}^n} \mathcal{L}\mathcal{Y}_{\kappa}(C_1,\ldots,C_n).$$ This definition is motivated by the original Lee-Yang theorem [31] and its various versions such as the so-called "circle theorem" ([6, Theorem 8.4], [23], [49]). For instance, the class of "complex Lee-Yang polynomials" related to the latter which was introduced and studied in [3] corresponds to the case when $C_i = \mathbb{D}$ , $i \in [n]$ , in Definition 6.1. Note also that by Corollary 1.10 the Lee-Yang polynomials with respect to $\{H_i\}_{i=1}^n$ , where $H_i = H$ , $i \in [n]$ , are precisely the polynomials that are non-zero complex constant multiples of real stable polynomials. We say that a linear operator $T: \mathbb{C}_{\kappa}[z_1,\ldots,z_n] \to \mathbb{C}[z_1,\ldots,z_n]$ preserves the $\kappa$ -Lee-Yang property with respect to $\{C_i\}_{i=1}^n$ if $T(f) \in \mathcal{LY}(C_1,\ldots,C_n) \cup \{0\}$ whenever $f \in \mathcal{LY}_{\kappa}(C_1,\ldots,C_n)$ . From our symbol theorems we can deduce a characterization of all such linear operators. We only display the case when T is non-degenerate in the sense that its image has dimension larger than 2.