Stable Multivariate Eulerian Polynomials

• Stable multivariate Eulerian polynomials are a class of polynomials that extend classical Eulerian polynomials by encoding descent statistics into several variables while retaining real stability.
• They establish a bridge between combinatorial enumeration and convex optimization via spectrahedral relaxations, enabling accurate approximation of rigidly convex sets.
• The methodology yields tight, sometimes exponentially-separated bounds for univariate extreme roots through refined eigenvector analysis and hyperbolicity certification.

Stable multivariate Eulerian polynomials are a class of polynomials that generalize the classical Eulerian polynomials by encoding descent structure and position statistics into several variables, while preserving strong analytic properties such as real stability or the real zero (RZ) property. They provide a natural interface between combinatorial enumeration, real algebraic geometry, and convex optimization via spectrahedral relaxations. The multivariate structure is crucial for constructing accurate global convex approximations of the "rigidly convex sets" (RCSs) defined by these polynomials, and—through refined linear algebraic analysis—yields tight, sometimes exponentially-separated, bounds for the extremal roots of univariate Eulerian polynomials.

1. Construction of Stable Multivariate Eulerian Polynomials

The multivariate Eulerian polynomials originate by lifting univariate Eulerian recurrence schemes into a multivariate, stable context. The classical Eulerian polynomial $A_n(x)$, enumerating permutations in $\mathfrak{S}_n$ by number of descents, obeys a recurrence that can be homogenized and extended: $$A_n^{\mathrm{h}}(x, y) = (x + y) A_{n-1}^{\mathrm{h}}(x, y) + xy\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y}\right)A_{n-1}^{\mathrm{h}}(x, y).$$ Iterative application of this operator yields a family of polynomials in $n+1$ variables once "auxiliary" homogenizing variables are added. Further, tagging positions (such as 'descent tops' and 'ascent tops') allows one to introduce vectors of variables $\mathbf{x}$ and $\mathbf{y}$: $$A_n(\mathbf{x}, \mathbf{y}) = \sum_{\sigma \in \mathfrak{S}_{n+1}} \prod_{i \in \mathcal{DT}(\sigma)} x_i \prod_{j \in \mathcal{AT}(\sigma)} y_j,$$ where $\mathcal{DT}(\sigma)$ and $\mathcal{AT}(\sigma)$ denote sets of descent and ascent tops of $\sigma$. The polynomial is then symmetric with respect to the underlying combinatorics.

By specializing some variables (for example, $\mathbf{y} = (1,\dots,1)$), one obtains a multivariate real zero polynomial $A_n(\mathbf{x}, \mathbf{1})$ that retains strong root-location properties in every direction (Nevado, 4 Jul 2025).

2. Rigidly Convex Sets and Spectrahedral Relaxations

Given a real zero (RZ) multivariate Eulerian polynomial $p(\mathbf{x})$, the associated rigidly convex set (RCS) is the closure of the connected component of $\mathbb{R}^n \setminus \{p=0\}$ containing the origin. These sets are, by construction, rigidly convex and central in the theory of hyperbolic polynomials.

To efficiently approximate these rigidly convex sets for analytic and optimization purposes, the paper constructs spectrahedral relaxations. This is done by associating to $p$ a monic symmetric linear matrix polynomial (MSLMP): $M_p(\mathbf{x}) = A_0 + \sum_{i=1}^n x_i A_i,$ where $A_0$ and $A_i$ are obtained as polynomial (cubic) functions of the coefficients of $p$, typically by evaluating a truncated logarithmic derivative ("L-form") on the moment matrix $M_{n,\leq 1}$ (i.e., monomials up to degree one).

The spectrahedron

$$S(p) = \{ \mathbf{a} \in \mathbb{R}^n : M_p(\mathbf{a}) \succeq 0 \}$$

contains the rigidly convex set of $p$. The size of $M_p$ is small (dimension $n+1$) and independent of the degree of $p$, making computations feasible at scale (Nevado, 4 Jul 2025).

3. Accuracy and Diagonal Analysis

The quality of the spectrahedral relaxation is assessed by restricting to the "diagonal"—that is, the argument $(x, \dots, x)$. In this direction, the multivariate Eulerian polynomial specializes to the univariate one: $A_n(x, \dots, x) = A_n(x).$ The spectrahedral relaxation then reduces to a pencil $M_n(x, \dots, x) = M_{n,0} + x M_{n, \Sigma}$ and one studies bounds for the smallest real root via the inequality $v^T(M_{n,0} + xM_{n,\Sigma})v \geq 0$ for carefully chosen (generalized) eigenvectors $v$. The corresponding $x_*$ provides a explicit bound for the extreme root of $A_n(x)$.

Through this approach, the obtained bound for the univariate extreme root is, for increasing $n$, tighter than previous bounds in the literature—an essential validation of the multivariate spectrahedral approach (Nevado, 4 Jul 2025).

4. Improved Bounds via Eigenvector Guessing and Asymptotics

Empirical and theoretical advances in (Nevado, 24 Jul 2025) show that naive ("constant entry") eigenvector guesses for diagonal relaxation yield improvements vanishing as $n \to \infty$. By analyzing numeric eigenvectors for diagaonal pencils and their combinatorial patterns, the paper identifies exponentially-decaying blocks in their entries.

A new constructed sequence of "linearizing" vectors for even $n=2m$,

$$\left\{ \left( y, \left(-2^{m-i}\right)_{i=3}^m, 0, \frac{1}{2}, (1)_{i=1}^m \right) \in \mathbb{R}^{n+1} \right\}_{n=1}^\infty,$$

substituted in the LMP linearization, yields a (certified) bound whose difference with the best univariate approach scales like $\sim (9/8)^m$, an exponential gap. This demonstrates that the multivariate information encoded by the stable polynomial structure is quantitatively significant for bounding roots and therefore for approximating the rigidly convex set (Nevado, 24 Jul 2025).

5. Stability, Hyperbolicity, and RZ Certification

A central premise is the real stability (hyperbolicity) of the multivariate Eulerian polynomials. For $A_n(\mathbf{x}, \mathbf{y})$ homogeneous and $A_n(\mathbf{x}, \mathbf{y})(0) \neq 0$, real stability combined with non-vanishing at the origin guarantees that dehomogenization (e.g., setting $\mathbf{y} = \mathbf{1}$) yields a real zero polynomial (Nevado, 6 Mar 2025). Hyperbolicity then ensures that every line $t \mapsto p(\mathbf{a} + t\mathbf{b})$ has only real roots.

This property is essential: the spectrahedral relaxation is guaranteed to contain the true RCS if and only if $p$ is RZ. Furthermore, the hyperbolicity theory links root location in specialized directions (diagonals) to the full multivariate stable structure (Nevado, 4 Jul 2025).

6. Summary Table: Key Aspects

Aspect	Main Feature	Papers
Construction	Tagged descent/ascent tops, symmetric multi-affine recurrence	(Nevado, 4 Jul 2025, Nevado, 24 Jul 2025)
Rigidly convex set	Defined by vanishing of stable/Eulerian RZ polynomial	(Nevado, 4 Jul 2025)
Spectrahedral relaxation	Small MSLMP, L-form, outer convex approximation	(Nevado, 4 Jul 2025, Nevado, 6 Mar 2025)
Diagonal analysis	Recovery of univariate, palindromic Eulerian polynomials; benchmark	(Nevado, 4 Jul 2025, Nevado, 24 Jul 2025)
Improved root bounds	Exponential separation from previous bounds via vector guessing	(Nevado, 24 Jul 2025)
Real stability/hyperbolicity	Certification of RZ property and validity of relaxation	(Nevado, 6 Mar 2025, Nevado, 4 Jul 2025)

• Spectrahedral relaxations and multivariate Eulerian polynomials: (Nevado, 4 Jul 2025)
• Improved bounds via eigenvector constructions: (Nevado, 24 Jul 2025)
• RZ property by dehomogenization: (Nevado, 6 Mar 2025)

Implications and Broader Context

The stable multivariate Eulerian polynomials introduced using this framework not only generalize classical combinatorial objects but underpin a new mechanism for producing global convex approximations (spectrahedra) for sets defined by RZ polynomials. Multivariate stable structure, certified by hyperbolicity and explicit dehomogenization arguments, ensures correctness of relaxations. The exponential separation in root bounds underlines the power of exploiting the full multivariate combinatorial and analytic data present in these polynomials.