Eulerian Rigidly Convex Sets

• Eulerian rigidly convex sets are convex subsets defined by real-zero multivariate Eulerian polynomials that encode complex combinatorial and algebraic properties.
• They allow the construction of efficient spectrahedral approximations using linear matrix inequalities, yielding tractable semidefinite programming relaxations.
• These methods provide exponential improvements in bounding extreme roots and offer practical insights for optimization in algebraic geometry and combinatorics.

An Eulerian rigidly convex set is a convex subset of Euclidean space defined as the rigidly convex set associated with a real zero multivariate Eulerian polynomial. This concept forms at the intersection of real algebraic geometry, spectral graph theory, and combinatorics, and plays a central role in the precise convex relaxation and structure theory of semialgebraic sets defined by highly symmetric or combinatorially structured polynomials. The distinctive algebraic and combinatorial features of Eulerian rigidly convex sets are leveraged in modern convex optimization through their tractable spectrahedral approximations, allowing for fine control over geometric and analytic invariants such as extreme roots and volume.

1. Algebraic and Combinatorial Foundations

Eulerian rigidly convex sets are defined via multivariate real zero (RZ) polynomials that generalize classical Eulerian polynomials. The classical Eulerian polynomial is

$$A_n(x) = \sum_{\sigma\in S_{n+1}} x^{\mathrm{des}(\sigma)}$$

where $\mathrm{des}(\sigma)$ counts descents in the permutation $\sigma$. In the multivariate setting, the polynomial is refined by "tagging" descent-top elements, giving

$$A_n(\mathbf{x},\mathbf{y}) = \sum_{\sigma\in S_{n+1}} \prod_{i\in \mathcal{DT}(\sigma)} x_i \prod_{j\in \mathcal{AT}(\sigma)} y_j,$$

where $\mathcal{DT}(\sigma)$ denotes descent tops and $\mathcal{AT}(\sigma)$ ascent tops; typically, the ascent-top variables are specialized to $1$ to yield an RZ polynomial $p(\mathbf{x})$. The associated rigidly convex set is the Euclidean closure of the connected component containing the origin in the set where $p(\mathbf{x}) > 0$: $$\mathrm{rcs}(p) = \left\{a \in \mathbb{R}^n : p(ta) \neq 0\ \forall t \in [0,1)\right\}.$$ This structure encodes refined combinatorial information (from the tagged descents) and admits a rich convex-geometric interpretation due to the RZ property, i.e., for any $w$, $t \mapsto p(u+tw)$ has only real zeros for any interior point $u$.

2. Spectrahedral Relaxations: Construction and Properties

For Eulerian rigidly convex sets, explicit outer convex approximations are obtained as spectrahedra through linear matrix inequalities (LMIs). Given a RZ polynomial $p$ with $p(0)\ne 0$, one constructs a monic symmetric linear matrix polynomial (MSLMP) $M_p(\mathbf{x})$ by entrywise application of an $L$-form (derived from the logarithmic expansion of $p$) onto a moment ("mold") matrix: $$M_{n,\leq 1} = (1, x_1, \ldots, x_n)^\top (1, x_1, \ldots, x_n).$$ The matrices $A_0 = L_p\circ M_{n,\leq 1}$ and $A_i = L_p\circ (x_i M_{n,\leq 1})$ assemble the pencil

$M_p(\mathbf{x}) = A_0 + \sum_{i=1}^n x_i A_i.$

The spectrahedral relaxation is then

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

and always contains the rigidly convex set: $\mathrm{rcs}(p) \subseteq S(p)$ (Nevado, 4 Jul 2025).

The construction depends only cubically on the number of variables and on the degree-three truncation of $p$, making it highly efficient for computational purposes (Nevado, 6 Mar 2025). This approach is naturally compatible with sums of squares (SOS) relaxations and moment-matrix techniques foundational in convex algebraic geometry.

3. Diagonal Accuracy and Root-Bounding Behavior

The diagonal restriction of the spectrahedral relaxation, where all variables are set equal, is a crucial analytic tool. Restricting $p$ and $M_p$ along the diagonal (i.e., $x_i = t$ for all $i$) recovers the univariate Eulerian polynomial. The spectrahedral determinant and its generalized eigenvalue problem yield accurate (often optimal) bounds on the extreme roots (particularly, the leftmost/root of largest modulus) of $A_n(t)$: $$A_n(t, \ldots, t) = A_n(t), \quad M_n(t) = M_{n,0} + t M_{n,\Sigma}.$$ Explicit computation shows that the diagonal relaxation matches or, with appropriate vector linearization, beats previously established analytic bounds from the literature—such as the Colucci or Sobolev estimates—for large $n$ (Nevado, 4 Jul 2025, Nevado, 6 Mar 2025).

Refinements using specially constructed vector sequences, e.g.

$$\{(y, (-2^{m-i})_{i=3}^m, (0,\tfrac{1}{2}), (1)_{i=1}^m)\}_{n=2m}$$

achieve an exponential asymptotic improvement in bounds over previous techniques, with differences growing like $(3/8)\cdot(9/8)^m$ as $n=2m \to\infty$ (Nevado, 24 Jul 2025). This establishes a rigorous exponential improvement in diagonal accuracy as $n$ increases.

4. Spectrahedral Relaxations: Linearization and Eigenvector Sequences

Obtaining practical, sharp bounds from spectrahedral relaxations in high dimensions requires efficient "linearization" strategies—using specially tailored vector sequences as proxies for eigenvectors. The optimal sequence for the diagonal problem displays nontrivial structure: starting with a free parameter $y$, it progresses through entries with stepwise transformations (e.g., by powers of $-2$), a distinct value of $1/2$, and a final tail of ones. Numerical experiments confirm that this vector sequence achieves an exponential separation between the multivariate and univariate bounds (Nevado, 24 Jul 2025).

This approach thereby reveals additional multivariate information about the geometry of the RZ polynomial's rigidly convex set that a strictly univariate analysis cannot access.

5. Relations to Symmetry, Convex Algebraic Geometry, and Valuations

Eulerian rigidly convex sets, when arising as K-invariant sets under group actions, and their spectrahedral relaxations enjoy powerful equivariance properties. The bijection arising from Kostant’s Convexity Theorem between invariant convex subsets in a polar representation and those invariant under the Weyl group preserves the class of rigidly convex sets (Bettiol et al., 6 Aug 2024). This compatibility guarantees that the symmetric algebraic and convex structure—crucial in the Eulerian setting—is maintained under symmetry reduction.

Additionally, in the functional analytic setting, intrinsic valuation theory classifies function-theoretic versions of rigidly convex sets by the Euler characteristic, volume, and polar volume components (Mussnig, 2018). These notions provide a deeper lens through which to interpret the geometric invariants and structure of Eulerian rigidly convex sets as determined by their defining polynomials.

6. Applications, Optimization, and Future Directions

Spectrahedral relaxations of Eulerian rigidly convex sets have a variety of applications:

• Optimization and SDP relaxations: The efficient spectrahedral approximation via small MSLMPs provides practical, theoretically controlled SDP relaxations for polynomial optimization and combinatorial enumeration (Nevado, 4 Jul 2025).
• Root bounds for combinatorial polynomials: The asymptotic sharpness in bounding the extreme roots of Eulerian polynomials has direct implications for the analysis of spectral and stability properties in both combinatorics and applied mathematics (Nevado, 6 Mar 2025, Nevado, 24 Jul 2025).
• Convexification of algebraic sets: Methods from theta bodies and SOS programming yield semidefinite representations or tight outer approximations for convex hulls of semialgebraic sets defined by Eulerian-type polynomials (1007.1191).

Future directions include identifying further stability-preserving multivariate liftings, optimizing the choice of eigenvector sequences to sharpen spectrahedral approximations, and extending these techniques to other families of combinatorial and hyperbolic polynomials. Deeper investigations into lateral and dual (as opposed to diagonal) relaxations, and the paper of the functional-analytic invariants of more general rigidly convex sets, are open areas with significant prospect for further theoretical and computational advances.