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

Theorem 1.1. Let κ ∈Nn and T : Cκ[z1, . . . , zn] →C[z1, . . . , zn] 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) = α(f)P, where α is a linear functional on Cκ[z1, . . . , zn] and P is a stable polynomial, or (b) GT (z, w) ∈H2n(C).

Theorem 1.2. Let κ ∈Nn and T : Rκ[z1, . . . , zn] →R[z1, . . . , zn] 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) = α(f)P + β(f)Q, where α, β : Rκ[z1, . . . , zn] →R are linear functionals and P, Q are real stable polynomials such that P ≪Q, or (b) GT (z, w) ∈H2n(R), or (c) GT (z, −w) ∈H2n(R). If T : K[z1, . . . , zn] →K[z1, . . . , zn], where K = R or C, is a linear operator we define its tra

Theorem 1.3. Let T : C[z1, . . . , zn] →C[z1, . . . , zn] 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) = α(f)P, where α is a linear functional on C[z1, . . . , zn] and P is a stable polynomial, or (b) GT (z, w) ∈H2n(C).

Theorem 1.4. Let T : R[z1, . . . , zn] →R[z1, . . . , zn] 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) = α(f)P + β(f)Q, where α, β : R[z1, . . . , zn] →R are linear functionals and P, Q are real stable polynomials such that P ≪Q, or (b) GT (z, w) ∈H2n(R), or (c) GT (z, −w) ∈H2n(R). 1.2. Fundamental Properties of Stable Polynomials. In this section we first review some basic facts and then prov

Lemma 2.1]).

Lemma 1.5. Let f ∈K[z1, . . . , zn], where K = R or C. Then f ∈Hn(K) if and only if f(λt + α) ∈H1(K) for all λ ∈Rn + and α ∈Rn. 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 proof.

Theorem 1.6 (Hurwitz’ theorem). Let D be a domain (open connected set) in Cn and suppose {fk}∞ k=1 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 ∈C : Im(z) > 0}n and fk(z1, . . . , zn) = f(z1, . . . , zi−1, µ + zi/k,

Lemma 1.5.

Lemma 1.7. Let K = R or C and f ∈Hn(K) be of degree dj in zj, 1 ≤j ≤n. Then for any 1 ≤i ≤n one has: (1) f(z1, . . . , zi−1, µ, zi+1, . . . , zn) ∈Hn−1(K) ∪{0} for µ ∈R; (2) f(z1, . . . , zi−1, λzi, zi+1, . . . , zn) ∈Hn(K) for λ > 0; (3) zdi i f(z1, . . . , zi−1, −z−1 i , zi+1, . . . , zn) ∈Hn(K); (4) f(z1, . . . , zi−1, zj, zi+1, . . . , zn) ∈Hn−1(K) for 1 ≤j ̸= i ≤n. We will now establish a series of results involving the notion of proper position introduced in Definition 1.1, compare with [9,

Lemma 1.8. Let f, g ∈R[z1, . . . , zn] \ {0}, set h = f + ig and suppose that f and g are not constant multiples of each other. The following are equivalent: (1) h ∈Hn(C), that is, g ≪f; (2) |h(z)| > |h(¯z)| for all z = (z1, . . . , zn) ∈Cn with Im(zj) > 0, 1 ≤j ≤n, where ¯z = (¯z1, . . . , ¯zn);

Theorem 1.9. Let f, g ∈R[z1, . . . , zn]. All non-zero polynomials in the space {αf + βg : α, β ∈R} are stable if and only if either f = g ≡0, f ≪g or g ≪f. Moreover, if g ≪f then Wj[g, f](x) ≤0 for all x ∈Rn and 1 ≤j ≤n, and if f ≪g then Wj[g, f](x) ≥0 for all x ∈Rn and 1 ≤j ≤n.

Corollary 1.10. If f, g ∈R[z1, . . . , zn] \ {0} are real stable polynomials such that f ≪g and g ≪f then f = αg for some α ∈R.

Lemma 2.1 (Lieb-Sokal). Let P(z) + wQ(z) ∈C[z1, . . . , zn, w] be stable. If the degree in the variable zj is at most one then the polynomial P(z) −∂Q(z) ∂zj is either identically zero or stable.

Lemma 2.2. Let T : C(1n)[z1, . . . , zn] →C[z1, . . . , zn] be a linear operator such that GT (z, w) := T h (z + w)[n]i = X S⊆[n] T  zS w[n]\S is stable, where z = (z1, . . . , zm) and w = (w1, . . . , wn). Then T preserves stability.

Theorem 2.3 (Grace-Walsh-Szeg¨o). Let f be a symmetric multi-affine polynomial in n complex variables, let C be an open or closed circular domain, and let ξ1, . . . , ξn 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 ξ ∈C such that f(ξ1, . . . , ξn) = f(ξ, . . . , ξ). (2.3) We give a new and self-contained proof of Theorem 2.3 in [6, §2].

Proposition 2.4. Let f ∈Cκ[z1, . . . , zn]. Then f is stable if and only if Π↑ κ(f) is stable.

Lemma 2.5. Let T : Cκ[z1, . . . , zn] →Cγ[z1, . . . , zn] be a linear operator. Then the symbol of the polarization of T is the polarization of the symbol of T , that is, GΠ(T ) = Π↑ γ⊕κ(GT ), where Π↑ γ⊕κ : Cγ⊕κ[z1, . . . , zn, w1, . . . , wn] →Cγ⊕κ MA .

Lemma 3.1. Let f ∈Cκ[z1, . . . , zn], where κ = (κ1, . . . , κn) ∈Nn, and W = (W1, . . . , Wn) ∈{ζ ∈C : Im(ζ) > 0}n. Then for all ǫ > 0 sufficiently small one has (z + W)κ + ǫf(z) ∈Hn(C), where (z + W)κ = (z1 + W1)κ1 · · · (zn + Wn)κn.

Lemma 3.2. Let V ⊆K[z1, . . . , zn] be a K-linear space, where K = R or C. (i) If K = R and every non-zero element of V is real stable then dim V ≤2. (ii) If K = C and every non-zero element of V is stable then dim V ≤1.

Proposition 4.1. For any n ∈N the following holds: Hn(C) ∩H− n (C) = CHn(R) := {cf : c ∈C, f ∈Hn(R)}.

Lemma 3.2 we conclude that the image of T is of dimension at most two. Thus we may assume that T [(z+W)κ] ∈Hn(C) for all W ∈{z ∈C : Im(z) > 0}n or T [(z −W)κ] ∈Hn(C) for all W ∈{z ∈C : Im(z) > 0}n. But this amounts to saying that T [(z + w)κ] ∈H2n(R) or T [(z −w)κ] ∈H2n(R), as claimed. □ 5. Soft-Core/Transcendental Classifications In this section we settle Theorems 1.3 and 1.4, that is, the transcendental char- acterizations of stability, respectively real stability preservers. Let K = C or R. A

Theorem 5.1. Let F(z, w) = P α∈Nn Pα(z)wα, where z = (z1, . . . , zn) and w = (w1, . . . , wn), be a formal power series in w with coefficients in K[z1, . . . , zn]. Then F(z, w) ∈H2n(K) if and only if X α≤β (β)αPα(z)wα ∈H2n(K) ∪{0} for all β ∈Nn. 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 K = R and K = C, we will only focus on the latter case. 5.1. Generalized Jensen Multipliers. For α, β ∈Nn l

Lemma 5.2. Let β ∈Nn. The linear operators on C[z1, . . . , zn] defined by zα 7→ J(α, β)zα, α ∈Nn, zα 7→ (β)αzα, α ∈Nn, preserve stability.

Lemma 5.3 (Sz´asz). Suppose that f(z) = 1+Pk i=1 aizi = Qk j=1(1+ξjz) is stable. Then k X j=1 |ξj|2 ≤3|a1|2 + 2|a2|.

Lemma 5.3: |f(z)| ≤exp  r|a1| + 3r2|a1|2 + 3r2|a2|  , |z| ≤r.

Lemma 5.4. Suppose that the polynomial f(z) = 1 + X |β|>0 a(β)zβ ∈C[z1, . . . , zn] is stable and let A =  3 n X i=1 |a(ei)| !2 + 2

Theorem 5.5. Suppose that the polynomial f(z) = 1 + X |β|>0 a(β)zβ ∈C[z1, . . . , zn] is stable and let B = 2n−1 √ 2e2 −e e −1 = 2n−1 · 2.0210 . . ., C = 6e2 n X i=1

Theorem 5.6. Let M ⊂Nn be a finite non-empty set and f(z) = P α a(α)zα ∈ C[z1, . . . , zn] be a stable polynomial with M(f) = M. Then there are constants B and C depending only on the coefficients a(α) with α ∈M2 such that max n |f(z)| : |zi| ≤r, 1 ≤i ≤n o ≤BeCr2 for all r ≥0. Moreover, B and C can be chosen so that they depend continuously on the aforementioned set of coefficients.

Lemma 6.1. Let {Ci}n i=1 be a family of circular domains, f ∈C[z1, . . . , zn] be of degree κ ∈Nn, and J ⊆[n] a (possibly empty) set such that Cj is the exterior of a disk whenever j ∈J. Denote by g the polynomial in the variables zj, j ∈J,

Lemma 6.2. Suppose that C1, . . . , Cn, D1, . . . , Dn are open circular domains and κ = (κ1, . . . , κn) ∈Nn. Then there are M¨obius transformations ζ 7→φi(ζ) = aiζ + bi ciζ + di , i ∈[n], (6.2) as in (6.1) such that the (invertible) linear transformation Φκ : Cκ[z1, . . . , zn] → Cκ[z1, . . . , zn] defined by Φκ(f)(z1, . . . , zn) = (c1z1 + d1)κ1 · · · (cnzn + dn)κnf(φ1(z1), . . . , φn(zn)) (6.3) restricts to a bijection between Nκ(C1, . . . , Cn) and Nκ(D1, . . . , Dn).

Theorem 6.3. Let κ ∈Nn and T : Cκ[z1, . . . , zn] →C[z1, . . . , zn] be a linear op- erator. Let further Ci, i ∈[n], be open circular domains given by Ci = φ−1 i (H), i ∈[n], where φi, i ∈[n], are M¨obius transformations as in (6.2) such that the cor- responding linear transformation Φκ defined in (6.3) restricts to a bijection between Nκ(H, . . . , H) and Nκ(C1, . . . , Cn) (cf. Lemma 6.2). Then T : Nκ(C1, . . . , Cn) →N(C1, . . . , Cn) ∪{0} if and only if either (a) T has range of dimension at

Theorem 6.4. Let κ ∈Nn, T : Cκ[z1, . . . , zn] →C[z1, . . . , zn] be a non-degenerate linear operator and Ci, i ∈[n], be open circular domains given by Ci = φ−1 i (H), i ∈[n], where φi, i ∈[n], are M¨obius transformations as in Theorem 6.3. Then T preserves the κ-Lee-Yang property with respect to {Ci}n i=1 if and only if either (a) the polynomial in 2n variables z1, . . . , zn, w1, . . . , wn given by T " n Y i=1 (aizi + bi)(ciwi + di) + (aiwi + bi)(cizi + di)
κi #

Theorem 7.1. Let κ ∈Nn and T : Cκ[z1, . . . , zn] →C[z1, . . . , zn] be a linear operator. If GT (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 : Cκ[z1, . . . , zn] →C[z1, . . . , zn] to preserve C1 × · · · × Cn-stability, where Ci, i ∈[n], are arbitrary (open) circular domains. For simplicity we only state here

Theorem 7.2. Let κ ∈Nn, T : Cκ[z1, . . . , zn] →C[z1, . . . , zn] a linear operator and C a convex closed circular domain given by C = φ−1(H), where φ is a M¨obius transformation as in (6.1). If T (az + b)(cw + d) + (aw + b)(cz + d)
κ is C-stable then T preserves C-stability.

Theorem 7.2 is just a constant multiple of T [(1+zw)κ] while for a closed half-plane bordering on the origin it is a constant multiple of GT (z, w) = T [(z + w)κ]. 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)κ is not strictly stable, cf. (i) in §8. 8. Further Directions To conclude we mention some of the most appealing cases where Problems 1–2 remain open: (i) Ωi