#Stable polynomials
Let \(\mathcal{H} \subset \setC\) denote the upper half-plane \(\{z \in \setC: \mathrm{im}(z) \gt{} 0 \}.\)
Definition (Stability).
A multivariate polynomial \(P \in \setC[z_1,\dotsc,z_n]\) is called stable if it does not vanish on \(\mathcal{H}^n.\) That is, \(P\) is stable if \[\zvec^* \in \mathcal{H}^n \implies P(\zvec^*) \neq 0.\] One can easily show that \(P \in \setR[z_1,\dotsc,z_n]\) is stable if and only if \(P(\alpha+\lambda t)=0\) has only real zeros for all \(\alpha \in \setR^n,\) \(\lambda \in \setR_+^n.\) A univariate polynomial with real coefficients is stable if and only if all its roots are real.
The Lee–Yang circle theorem is a foundational circular-domain analogue. For a finite ferromagnetic Ising model in a uniform external field, every zero of the partition function, viewed as a polynomial in the fugacity, lies on the unit circle [LY52]. A Möbius transformation between the disk and the upper half-plane transports such zero-free regions into the language of stability; this viewpoint underlies the modern Lee–Yang program [BB09].
Definition (Strongly Rayleigh, [BBL08]).
A measure \(\mu\) on the set of subsets of \([n]\) is called strongly Rayleigh if \[\sum_{S \subseteq [n]} \mu(S) \prod_{s\in S} x_s\] is real stable.
#Operators preserving stability
Theorem (Stability preserving operators).
The following operations on polynomials preserve stability. Let \(p,q \in \setR[z_1,\dotsc,z_n]\) be stable polynomials. Then the following are also stable.
\(p\cdot q\) (Product)
\(\partial_{z_j} p\) (Differentiation)
\(z_j \partial_{z_j} p\) (Degree-preserving differentiation)
\(p(z_1,\dotsc,z_{j-1},r,z_{j+1},\dotsc,z_n)\) (Real specialization)
\(p(z_1,\dotsc,z_{j-1},z_1,z_{j+1},\dotsc,z_n)\) (Projection)
\(z_j^d p(z_1,\dotsc,z_{j-1},-1/z_{j},z_{j+1},\dotsc,z_n)\) (Inversion)
Here, \(d\) is the degree of \(z_j\) in \(p.\)
#Borcea–Brändén symbols
The Borcea–Brändén classification packages a linear operator into a polynomial, or into an entire power series, called its symbol. This is often the most practical way to prove that an operator preserves stability: instead of testing the operator on all stable inputs, one tests one universal object. For real stability one must allow both signs. For instance, \(f(z)\mapsto f(-z)\) preserves real-rootedness, but it is detected by the minus symbol rather than the plus symbol.
Definition (Algebraic symbols).
Let \(\mathbf{d}=(d_1,\dotsc,d_n) \in \setN^n,\) and let \(\setR_{\leq \mathbf{d}}[z_1,\dotsc,z_n]\) denote the polynomials whose degree in \(z_j\) is at most \(d_j.\) For a linear operator \[T:\setR_{\leq \mathbf{d}}[z_1,\dotsc,z_n]\to \setR[x_1,\dotsc,x_m],\] define the two algebraic symbols of \(T\) by \[G_T^{+}(\xvec,\mathbf{w}) \coloneqq T\left( \prod_{j=1}^n (z_j+w_j)^{d_j} \right), \qquad G_T^{-}(\xvec,\mathbf{w}) \coloneqq T\left( \prod_{j=1}^n (z_j-w_j)^{d_j} \right).\] Here \(T\) acts on the \(z\)-variables and the \(w_j\)’s are auxiliary variables.
Theorem (Borcea–Brändén finite-degree classification, see [BB09]).
Let \(T:\setR_{\leq \mathbf{d}}[z_1,\dotsc,z_n]\to \setR[x_1,\dotsc,x_m]\) be linear. Then \(T\) sends every real stable polynomial to a real stable polynomial or to zero if and only if one of the following holds.
\(G_T^{+}\) is real stable.
\(G_T^{-}\) is real stable.
The range of \(T\) has dimension at most two and can be written as \(T(f)=\alpha(f)P+\beta(f)Q,\) where \(\alpha,\beta\) are real linear functionals and \(P,Q\) are real stable polynomials in proper position. In one common convention this means that \(Q+iP\) is stable.
For operators over complex coefficients, the corresponding stability-preserver classification has only the plus symbol, apart from the rank-one exceptional case. The extra sign in the real theorem records the distinction between preserving the upper half-plane and preserving real-rootedness.
Example (Two elementary symbols).
For \(T=\partial_{z_i},\) the plus symbol is \[G_T^{+}(\xvec,\mathbf{w}) = d_i(x_i+w_i)^{d_i-1} \prod_{j\neq i}(x_j+w_j)^{d_j},\] which is stable. This recovers the stability-preserving property of partial differentiation.
For the univariate reflection \(T(f)(x)=f(-x)\) on \(\setR_{\leq d}[z],\) the plus symbol \((w-x)^d\) is not stable, but the minus symbol \[G_T^{-}(x,w)=(-x-w)^d\] is stable. Hence the sign alternative is genuinely needed for real stability preservers.
Definition (Laguerre–Pólya class and transcendental symbol).
Let \(\mathcal{LP}_r\) be the Laguerre–Pólya class of entire functions in \(r\) variables that are limits, uniformly on compact sets, of real stable polynomials. This is the multivariate Laguerre–Pólya class.
For a linear operator \[T:\setR[z_1,\dotsc,z_n]\to \setR[x_1,\dotsc,x_m],\] its transcendental symbol is the formal power series \[\overline{G}_T(\xvec,\mathbf{w}) \coloneqq T\!\left(e^{-\zvec\cdot\mathbf{w}}\right) = \sum_{\alpha\in\setN^n} (-1)^{|\alpha|} T(\zvec^\alpha) \frac{\mathbf{w}^{\alpha}}{\alpha!}.\]
Theorem (Borcea–Brändén transcendental classification, see [BB09]).
Up to the same rank-at-most-two exception as above, a linear operator \[T:\setR[z_1,\dotsc,z_n]\to \setR[x_1,\dotsc,x_m]\] preserves real stability if and only if \[\overline{G}_T(\xvec,\mathbf{w})\in \mathcal{LP}_{m+n} \qquad\text{or}\qquad \overline{G}_T(\xvec,-\mathbf{w})\in \mathcal{LP}_{m+n}.\]
This is the infinite-degree analogue of the algebraic-symbol test. It also explains why the classical multiplier sequence theorem has an exponential generating function. If \[T_\gamma(z^k)=\gamma_k z^k,\] then \[\overline{G}_{T_\gamma}(x,w) = \sum_{k\geq 0}\gamma_k\frac{(-xw)^k}{k!} = \Phi_\gamma(-xw).\] Thus the Pólya–Schur condition on \(\Phi_\gamma\) is the diagonal-operator case of the Borcea–Brändén symbol criterion. Compare this with Pólya frequency sequences: PF sequences are ordinary coefficient sequences controlled by Toeplitz total positivity, whereas multiplier sequences are diagonal operators controlled by exponential symbols.
M.-J. Ding and B.-X. Zhu give a systematic treatment of stability for combinatorial polynomials [DZ21]. They apply Borcea–Brändén type characterizations to real and Hurwitz stability, Turán expressions, alternating-runs polynomials, and several interlacing and alternatingly-increasing phenomena.
H. Fang and B. Ma introduce Gårding polynomials, a class strictly containing real stable polynomials [FM26]. The definition is phrased in terms of positivity regions invariant under translation by positive vectors; they prove equivalent formulations by polarization and by a recursive criterion using partial derivatives. Multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations have ultra-log-concave coefficient sequences.
Their matroid applications cover cycle matroids of series–parallel networks, uniform matroids, matroids obtained from a uniform matroid by removing one basis, and every matroid on at most six elements. For these classes, the spanning-set, cospanning-set, and basis generating polynomials are Gårding. All of these polynomials, as well as the corresponding independent-set generating polynomials, are Rayleigh, and their univariate specializations satisfy ultra-log-concavity and Mason-type inequalities [Thm. 1.3, FM26].
#Capacity and coefficient bounds
For a homogeneous polynomial \(p\in\setR[x_1,\dotsc,x_n]\) of degree \(n\) with nonnegative coefficients, define its capacity by \[\operatorname{Cap}(p) \coloneqq \inf_{x_1,\dotsc,x_n\gt{}0}\frac{p(x_1,\dotsc,x_n)}{x_1\dotsm x_n}.\]
Theorem (Gurvits, [Thm. 2.4 and Cor. 2.5, Gur08]).
If \(p\) is real stable, then \[\frac{\partial^n p}{\partial x_1\dotsm\partial x_n}(0) \geq \frac{n!}{n^n}\operatorname{Cap}(p).\]
This turns stability into a coefficient lower bound and behaves well under successive differentiation. For a doubly stochastic matrix \(A=(a_{ij}),\) the polynomial \[p_A(\xvec)=\prod_{i=1}^n\left(\sum_{j=1}^n a_{ij}x_j\right)\] is real stable and has capacity one, while the coefficient of \(x_1\dotsm x_n\) is \(\operatorname{per}(A).\) The theorem therefore gives the van der Waerden bound \(\operatorname{per}(A)\geq n!/n^n.\)
#Multi-affine stability
A polynomial \(P(z_1,\dotsc,z_n)\) is multi-affine if the degree in each variable is at most \(1.\)
Theorem (See [Br07]).
A homogeneous multi-affine (real) polynomial \(f(x_1,\dotsc,x_n)\) is real stable if and only if \[\frac{\partial f}{\partial x_i} \cdot \frac{\partial f}{\partial x_j} - \frac{\partial^2 f}{\partial x_i \partial x_j} \cdot f \geq 0\] whenever \(x_1,x_2,\dotsc,x_n \in \setR.\)
A polynomial satisfying this inequality is said to have the strongly Rayleigh property. The weak Rayleigh property holds if the above inequality is true whenever the variables are positive real numbers.
Theorem (Grace–Walsh–Szegő, see [Gra02, Sze22, Wal22, RS02]).
Let \(P(z_1,\dotsc,z_n)\) be a symmetric multi-affine polynomial. Then \(P\) is stable if and only if the univariate diagonal specialization \(P(t,\dotsc,t)\) is stable. If \(P\) has real coefficients, this is equivalent to the univariate specialization \(P(t,\dotsc,t)\) being real-rooted.
Definition (Apolar polynomials).
Let \[F(z)=\sum_{k=0}^n \binom{n}{k} a_k z^k \qquad\text{and}\qquad G(z)=\sum_{k=0}^n \binom{n}{k} b_k z^k.\] The polynomials \(F\) and \(G\) are apolar if \[\sum_{k=0}^n (-1)^k \binom{n}{k} a_k b_{n-k}=0.\]
Theorem (Grace’s apolarity theorem, see [Gra02, Sze22, RS02]).
Let \(C\) be a circular domain, and let \(F\) and \(G\) be apolar polynomials of degree \(n.\) If all zeros of \(F\) lie in \(C,\) then \(G\) has at least one zero in \(C.\)
A useful way to reduce questions to the multi-affine case is polarization. If \(0 \leq k \leq d_j,\) set \[\operatorname{Pol}_{d_j}(z_j^k) \coloneqq \binom{d_j}{k}^{-1} \elementaryE_k(z_{j,1},\dotsc,z_{j,d_j}).\] For monomials, define \[\operatorname{Pol}_{\mathbf{d}}(\zvec^\alpha) \coloneqq \prod_{j=1}^n \operatorname{Pol}_{d_j}(z_j^{\alpha_j}),\] and extend linearly to polynomials. This gives the polarization \(\operatorname{Pol}_{\mathbf{d}}(P)\) of a polynomial whose degree in \(z_j\) is at most \(d_j.\)
Theorem (Polarization, see [BB09]).
Let \(P \in \setC[z_1,\dotsc,z_n]\) have degree at most \(d_j\) in the variable \(z_j.\) Then \(P\) is stable if and only if \(\operatorname{Pol}_{\mathbf{d}}(P)\) is stable.
#Jump systems and matroids
For \(P = \sum_{\alpha} c_\alpha \zvec^\alpha,\) let \(\supp(P) \coloneqq \{\alpha : c_\alpha \neq 0\}.\)
Theorem (See [Br07]).
If \(P\) is stable, then \(\supp(P)\) is a jump system.
Theorem (See [COSW04]).
If \(P\) is homogeneous and multi-affine stable, then its support is the set of bases of a matroid. See also [Cor. 3.4, Br07].
The converse is in general not true but some classes of matroids always have a basis-generating polynomial which is stable. For example, regular matroids (which include graphic matroids), uniform matroids and a subclass of transversal matroids (see [COSW04]).
Deletion and contraction on matroids preserve stability of basis-generating polynomials.
For a connected graph \(G\) on vertex set \(V,\) the spanning-tree degree enumerator \[\sum_T \prod_{v \in V} x_v^{\deg_T(v)},\] where the sum is over all spanning trees \(T\) of \(G,\) is real stable if and only if \(G\) is distance-hereditary [CPP23].
Theorem (See [BH20]).
Let \(P \in \setR[z_1,\dotsc,z_n]\) be homogeneous, real stable, and have nonnegative coefficients. Then \(P\) is Lorentzian.
#Determinantal stability and the Lax theorem
Theorem (Borcea–Brändén, [Prop. 1.12, BB10]; see also [Prop. 2.4, BB08]).
Let \(A,B_1,\dotsc,B_n\) be Hermitian \(m\times m\) matrices, and suppose that \(B_1,\dotsc,B_n\) are positive semidefinite. Then \[\det(A+z_1B_1+\dotsb+z_nB_n)\] is stable, in fact real stable, or identically zero.
For real values of the variables, the matrix pencil is Hermitian and its determinant is real, so the determinant polynomial has real coefficients. The proof rests on the interaction between the upper half-plane and positive semidefiniteness. Suppose that all \(\operatorname{Im}(z_i)\gt{}0\) and that \(M=A+\sum_i z_iB_i\) is singular. For a nonzero vector \(v\) in its kernel, \[0=\operatorname{Im}(v^*Mv) =\sum_i \operatorname{Im}(z_i)v^*B_iv.\] Every summand is nonnegative, so \(B_iv=0\) for every \(i,\) and then \(Av=0.\) Thus \(v\) lies in the kernel of every matrix in the pencil, making the determinant identically zero. Hence a nonzero determinant of this form cannot vanish when all variables lie in the upper half-plane.
#Mixed characteristic polynomials and barriers
For positive semidefinite Hermitian matrices \(A_1,\dotsc,A_m,\) define \[\mu[A_1,\dotsc,A_m](x) \coloneqq \left. \prod_{i=1}^m(1-\partial_{z_i}) \det\left(xI+\sum_{i=1}^m z_iA_i\right) \right|_{z_1=\dotsb=z_m=0}.\] This is the mixed characteristic polynomial .
Theorem (Marcus–Spielman–Srivastava, [Cor. 4.4 and Thm. 5.1, MSS15]).
The polynomial \(\mu[A_1,\dotsc,A_m](x)\) is real-rooted. If additionally \[\sum_{i=1}^m A_i=I \qquad\text{and}\qquad \operatorname{tr}(A_i)\leq\epsilon\quad\text{for every $i$},\] then its largest zero is at most \((1+\sqrt{\epsilon})^2.\)
Real-rootedness follows by applying the stability-preserving operators \(1-\partial_{z_i}\) to the determinantal polynomial. The root bound is proved by tracking logarithmic-derivative barrier functions as these operators are applied. Together with interlacing families, this is the main polynomial tool in the solution of the Kadison–Singer problem.
Theorem (Borcea–Brändén, [Thm. 1.13 and Cor. 6.7, BB10]).
Let \(P(z,w)\in\setR[z,w]\) have degree \(d.\) Then \(P\) is real stable if and only if there are real symmetric \(d\times d\) matrices \(A,B,C,\) with \(B\) and \(C\) positive semidefinite, such that \[P(z,w)=\pm\det(A+zB+wC).\]
This converse is the two-variable stable-polynomial form of the classical Lax theorem. In its ternary form, the theorem states that if a homogeneous \(h(x,y,t)\in\setR[x,y,t]\) of degree \(d\) is hyperbolic with respect to \((1,0,0),\) then \[h(x,y,t)=h(1,0,0)\det(xI+yB+tC)\] for real symmetric \(d\times d\) matrices \(B,C.\) The Lax conjecture was proved by J. W. Helton and V. Vinnikov, and by A. Lewis, P. Parrilo, and M. Ramana [HV07, LPR05].
To obtain the bivariate converse, one homogenizes \(P\) to \[P_H(z,w,t)\coloneqq t^dP(z/t,w/t).\] Stability makes \(P_H\) hyperbolic with respect to every direction \((a,b,0)\) with \(a,b\gt{}0.\) Applying the ternary Lax theorem and using all these directions forces the matrices multiplying \(z\) and \(w\) to be positive semidefinite, which gives the representation above.
The bivariate conclusion does not extend to a polynomial-level statement in arbitrarily many variables. P. Brändén used the stable basis-generating polynomial \(h_{V_8}\) of the Vámos matroid to construct the real-zero polynomial \(p(\xvec)=h_{V_8}(\mathbf{1}+\xvec),\) for which no positive power \(p^N\) has a determinantal representation [Thm. 3.3, Br11]. Consequently, \(h_{V_8}\) and its powers need not themselves have the analogous homogeneous determinant representation. This does not disprove the generalized cone-level Lax conjecture: that conjecture asks only whether every hyperbolicity cone is spectrahedral, possibly through a different defining polynomial, and remains open.
Example (The monotone column permanent conjecture).
Let \(A\) be a real \(n\times n\)-matrix in which entries along columns decrease. Let \(Z_n = \mathrm{diag}(z_1,\dotsc,z_n)\) be a diagonal matrix with indeterminates, and let \(J_n\) be the matrix with all entries equal to \(1.\) Then \[\mathrm{per}(J_nZ_n + A) = \mathrm{per}\left( (z_j + a_{ij})_{1 \leq i,j \leq n} \right)\] is stable, see [BHVW11]. By noting that the Eulerian polynomials are generated by excedances, they give a new proof that (a multivariate version of) Eulerian polynomials are stable.
Example (Multivariate Eulerian polynomials).
P. Brändén (see [Thm. 2.5, HV12]) proved stability of the following multivariate refinement of Eulerian polynomials. Set \[\widetilde A_n(\xvec,\yvec) = \sum_{\pi \in \symS_n} \prod_{\substack{1\leq i\lt{}n\\ \pi_i \gt \pi_{i+1}}} x_{\pi_i} \prod_{\substack{1\leq i\lt{}n\\ \pi_i \lt \pi_{i+1}}} y_{\pi_{i+1}}.\] Thus descent tops receive \(x\)-variables and ascent tops receive \(y\)-variables. The proof uses a stability-preserving operator. If \[\partial = \sum_{i=1}^n \frac{\partial}{\partial x_i} + \sum_{i=1}^n \frac{\partial}{\partial y_i},\] then \[T_{n+1} \coloneqq (x_{n+1}+y_{n+1}) + x_{n+1}y_{n+1}\partial\] preserves real stability on multi-affine polynomials, by the Borcea–Brändén symbol criterion. Equivalently, if \[U=\prod_{i=1}^n (x_i+u_i)(y_i+v_i),\] then the algebraic symbol \[T_{n+1}(U) = \left( x_{n+1}+y_{n+1} +x_{n+1}y_{n+1} \sum_{i=1}^n \left(\frac{1}{x_i+u_i}+\frac{1}{y_i+v_i}\right) \right)U\] is stable. The recurrence \[\widetilde A_{n+1}(\xvec,\yvec) = T_{n+1}\widetilde A_n(\xvec,\yvec)\] then proves stability by induction from \(\widetilde A_1=1.\) Specializing \(y_i=1\) and then \(x_i=t\) gives \[t\,\widetilde A_n(t,\dotsc,t;1,\dotsc,1)=A_n(t),\] so the classical Eulerian polynomial has only real roots.
In fact, J. Haglund and M. Visontai also treat Stirling permutations, where plateaus occur [HV12]. If \(Q_n\) is the set of Stirling permutations and \(D(\sigma),A(\sigma),P(\sigma)\) are the descent-top, ascent-top, and plateau sets, then \[C_n(\xvec,\yvec,\zvec) = \sum_{\sigma \in Q_n} \prod_{i\in D(\sigma)} x_{\sigma_i} \prod_{i\in A(\sigma)} y_{\sigma_i} \prod_{i\in P(\sigma)} z_{\sigma_i}\] is stable. Specializing \(y_i=z_i=1\) recovers real-rootedness of the second-order Eulerian polynomials.
Example (Multivariate matching polynomials).
The multivariate matching polynomial is real stable; see [BB09, BB09]. J. D. Leake and N. R. Ryder distinguish two types of multivariate matching polynomials, recording either vertex labels or edge labels in the matchings [LR19]: \[\begin{aligned} \mu_V(G;x_1,\dotsc,x_n) & \coloneqq \sum_{\substack{M \subseteq E(G) \\ \text{$M$ matching}}} \prod_{uv \in M} -x_u x_v, \\ \mu_E(G;x_1,\dotsc,x_n) & \coloneqq \sum_{\substack{M \subseteq E(G) \\ \text{$M$ matching}}} \prod_{e \in M} x_e. \end{aligned}\]
The multi-affine vertex matching polynomial \(\mu_V(G;x_1,\dotsc,x_n)\) is real stable, see [BB09].
D.G. Wagner’s survey [Wag09] is a useful reference for the theory and applications of multivariate stable polynomials, including the Borcea–Brändén classification of stability preservers, matrix applications, and probabilistic applications.
Theorem (Borcea–Brändén–Liggett, [Thm. 4.9, BBL08]).
Let \(\mu\) be a probability measure on subsets of \([n]\) whose generating polynomial \[\sum_{S\subseteq[n]}\mu(S)\prod_{i\in S}x_i\] is real stable. Then every measure obtained from \(\mu\) by applying positive external fields is negatively associated: increasing functions depending on disjoint coordinate sets have nonpositive covariance.
In fact the conclusion is the stronger property CNA+: it persists after conditioning, projection, and positive external fields. Determinantal measures induced by positive contractions are strongly Rayleigh; this includes product measures and uniform random spanning-tree measures [Prop. 3.5, BBL08].
#Same phase stability
J. D. Leake and N. R. Ryder introduced the following notion, which is strictly weaker than stability [LR19].
Definition (Same-phase stability).
A polynomial \(P(z_1,\dotsc,z_n) \in \setR[z_1,\dotsc,z_n]\) is said to be same-phase stable if for every \(\lambda \in \setR_+^n,\) the univariate polynomial \(P(\lambda_1 t,\lambda_2 t,\dotsc,\lambda_n t) \in \setR[t]\) is real-rooted.
A family of polynomials \(\{p_j(\xvec) \}_{j=1}^m\) is same-phase compatible if for every vector of positive numbers \(\lambda,\) the univariate polynomials \(\{p_j(\lambda t) \}_{j=1}^m\) are compatible.
Let \(P, P_0, \dotsc, P_m\) be polynomials such that \(P = P_0 + \sum_{j=1}^m z_{i_j} P_{j}.\) This is a proper splitting of \(P\) if none of the polynomials \(P_j\) depends on the variables \(\{z_{i_1}, \dotsc, z_{i_m} \}.\)
Theorem (See [LR19]).
Suppose \(P\) is a multi-affine polynomial with nonnegative coefficients. Then the following are equivalent:
\(P\) is same-phase stable;
For any proper splitting, \(P = P_0 + \sum_{j=1}^m z_{i_j} P_{j},\) the polynomials \(P_0, z_{i_1} P_{1}, \dotsc, z_{i_m} P_{m}\) are same-phase compatible.
Some proper splitting \(P = P_0 + \sum_{j=1}^m z_{i_j} P_{j}\) such that \(P_0, z_{i_1} P_{1}, \dotsc, z_{i_m} P_{m}\) are same-phase compatible.
The multi-affine edge matching polynomial \(\mu_E(G;x_1,\dotsc,x_n)\) is same-phase stable [Cor. 3.2, LR19].
Let \(G\) be a graph on the vertex set \([n],\) and define the multivariate independence polynomial \[I(G;x_1,\dotsc,x_n) \coloneqq \sum_{\substack{A \subseteq V(G) \\ A \text{ independent}}} \prod_{v \in A} x_v.\] Leake and Ryder prove that \(I(G;x_1,\dotsc,x_n)\) is same-phase stable if and only if \(G\) is claw-free [LR19]. This is a reinterpretation of Engström’s weighted result [Eng07], but Leake and Ryder give a much sleeker proof, and the if-and-only-if condition explains that the claw-free requirement is sharp.
Example (Multivariate Eulerian polynomials II).
In joint work with O. Nabawanda, we show that the following generalization of the Eulerian polynomial is same-phase stable, but not stable [AN21]: \[\tilde{A}_n(\xvec) = \sum_{\pi \in \symS_n} \prod_{\pi_i \gt \pi_{i+1}} x_{i}.\] Note that we now keep track of the descent index, rather than the descent bottom value. Moreover, we showed that \[\tilde{A}_{n-1}(\lambda_1 t, \dotsc, \lambda_{n-1} t) \interl \tilde{A}_n(\lambda_1 t, \dotsc, \lambda_n t)\] for all positive real numbers \(\lambda_1,\dotsc,\lambda_n.\)
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