Generalized Interlacing Families of Polynomials

• Generalized Interlacing Families of Polynomials are defined as collections of real-rooted polynomials arranged in a rooted tree where each node’s polynomial interlaces with its children, ensuring stability.
• The framework employs methods like convex combinations, finite free convolutions, and matrix operations to preserve root location properties, aiding in precise spectral and error bound estimates.
• Applications span spectral graph theory, deterministic approximation algorithms, and construction of Ramanujan graphs, illustrating practical insights across combinatorial and numerical linear algebra.

Generalized interlacing families of polynomials constitute a broad, unifying framework that extends the classical concept of polynomial interlacing to accommodate complex combinatorial, algebraic, and analytical settings. This apparatus systematically structures families of real-rooted polynomials (often arising in random matrix theory, graph spectra, approximation algorithms, orthogonal polynomials, and combinatorial enumeration) so that root location bounds, existence proofs, stability results, and error estimates for associated operators can be deduced via machinery generalizing the standard Sturm and orthogonal polynomial theory. Generalized interlacing families capture both convex combinations and more sophisticated algebraic operations (e.g., finite free convolutions, matrix products, section operators, and nonlinear recurrence systems) on polynomial families while preserving intrinsic root-interlacing and total positivity properties.

1. Formal Definitions and Core Properties

A generalized interlacing family is defined in the rooted tree framework, where each node is labeled by a real-rooted univariate polynomial, and the children of any internal node correspond to polynomials whose degrees are either equal to or one less than the root’s degree. Crucially, at each branching step, the collection must admit a common interlacer (i.e., a single real-rooted polynomial whose roots alternate with those of every branch polynomial) (Cai et al., 7 Dec 2025). The root polynomial and its convex combination of child polynomials preserve real-rootedness under expectation or summation, a property exploited for probabilistic existence proofs such as the construction of bipartite Ramanujan graphs (Marcus et al., 2015).

The central existence theorem asserts that in any generalized interlacing family, there exists at least one leaf whose smallest root is at least as large as the smallest root of the root polynomial. For families indexed by products of parameter spaces (e.g., swap choices, permutation blocks), averaging over all configurations yields an expected polynomial whose root locations can be leveraged for extremal existence via interlacing (Cai et al., 7 Dec 2025, Marcus et al., 2013).

2. Algebraic Constructions: Matrix Operations, Free Convolutions, Recurrences

Generalized interlacing phenomena arise naturally in matrix constructions and convolution algebras. Total positivity—preservation of non-negative determinants of minors in infinite matrices assembled from formal power series—characterizes fully interlacing families beyond the classical sequence setting. The property is stable under matrix products, reverse-diagonal flips, and Veronese section operators: if two matrices of power series are fully interlacing, their product remains fully interlacing, and the Veronese section operation preserves full interlacing as well (Athanasiadis et al., 19 Apr 2024).

Finite free additive convolution extends linear averaging to polynomial convolution operations indexed by matrix dimensions. For polynomials arising as characteristic polynomials of random matrices, expected polynomials under random permutations or Haar measures are shown to be real-rooted—with root location bounds given in terms of Cauchy transforms and mesh estimates (Marcus et al., 2015). The preservation of interlacing under convolution, sectioning, and other operations enables the derivation of nontrivial extremal graph spectra and tight error bounds in algorithms such as CUR decompositions (Cai et al., 7 Dec 2025).

3. Combinatorial, Orthogonal, and Topological Instances

Numerous combinatorial polynomial families demonstrate generalized interlacing:

• Dyck path pattern-count generating polynomials $W_{n,k}(x)$ (enumerating certain path statistics) have been shown to form Sturm sequences and Sturm-unimodal families via recurrences meeting appropriate sign conditions (Wang et al., 2023).
• Reflexive polytope Ehrhart polynomials and graph-polytopal constructions yield families with zeros symmetric and interlaced along critical lines in the complex plane (e.g., $\Re(z)=-1/2$), generalizing local Riemann hypothesis results and connecting to Mellin transforms of classical orthogonal systems (Higashitani et al., 2016).
• q-hypergeometric families such as the little q-Jacobi, q-Bessel, and Stieltjes-Wigert polynomials exhibit strict root interlacing under mesh constraints, contiguous relations, and parameter shifts, with the entire interlacing structure determined by sign-controlled q-difference equations and Jordan–Tóokos type lemmas (Martinez-Finkelshtein et al., 5 Jun 2025).

In more advanced settings, topological recursion generates polynomial families whose real-rootedness and interlacing persist beyond classical cases, e.g., deformed monotone Hurwitz and dessin d’enfant polynomials (Coulter et al., 2023).

4. Applications in Spectral Bounds, Approximation, and Algorithmic Design

Generalized interlacing families underpin several recent advances in deterministic approximation algorithms:

• In CUR and row-subset selection for matrix recovery, one constructs generalized interlacing trees on polytope-indexed polynomial families yielding spectral norm error bounds in terms of extremal roots. The existence lemma ensures the applicability of these bounds to explicit, efficiently findable subsets (Cai et al., 7 Dec 2025).
• The Marcus–Spielman–Srivastava program for solving Kadison-Singer, paving, and Weaver’s conjecture utilizes interlacing families of mixed characteristic polynomials to guarantee operator-norm partition bounds in random matrix and operator decomposition problems (Marcus et al., 2013).
• Interlacing machinery drives constructive proofs of existence for Ramanujan graphs, root location bounds for random matching unions, and extremal geometric configurations in combinatorial algebraic settings (Marcus et al., 2015).

5. Generalized Interlacing in Orthogonal, Nonclassical, and Mixed Polynomial Sequences

Generalized interlacing theorems extend not only within a single polynomial family but also across pairs or triples of different sequences linked by mixed recurrence or functional relations. For example, a linear factor $(x-A)g_n(x)$ constructed from an interlacing triple $\{p_n,g_n,q_{n+1}\}$ yields new interlacing families interspersed with prescribed points (e.g., parameter shift locations) (Jooste et al., 7 Dec 2024). Applications span Meixner–Pollaczek, Pseudo-Jacobi, and Continuous Hahn polynomials, with explicit formulas for interlacing points and functional relations dictating root alternation.

A significant direction concerns the characterization of all polynomial pairs (or parameter shifts) admitting such mixed interlacing via a linear or rational transformation, encompassing classical and nonclassical orthogonal systems, quasi-orthogonality, and families parametrized by spectral curves or topological recursion.

6. Unifying Frameworks and Outlook

The algebraic language of generalized interlacing families—whether via expectation, convolution, matrix product, or functional recurrence—unifies stability, total positivity, and root location control for combinatorial, analytic, and algorithmic polynomial families. This machinery has absorbed and extended concepts from Sturm sequences, orthogonal systems, real-stable polynomials, total positivity matrices, and hypergeometric constructions, systematically yielding real-rootedness, interlacing, and unimodality properties.

Future advances include:

• Generalizing the interlacing framework to multivariate polynomials, matrix-valued systems, and noncommutative settings.
• Characterizing the full class of operations, deformations, and parameter shifts preserving generalized interlacing (especially in q-orthogonal, topological, and geometric families).
• Algorithmic utilization in deterministic selection, approximation, and extremal configuration problems in numerical linear algebra, combinatorics, and theoretical computer science.
• Exploring connections to free probability, operator algebras, and spectral graph theory, where convolution-based interlacing provides analytic control.

Generalized interlacing families thus represent a flexible and technically robust substrate for much of modern algebraic combinatorics, spectral theory, and applied mathematics, enabling deep structural insight and effective analytical tools (Marcus et al., 2015, Athanasiadis et al., 19 Apr 2024, Coulter et al., 2023, Marcus et al., 2013, Higashitani et al., 2016, Cai et al., 7 Dec 2025, Jooste et al., 7 Dec 2024, Wang et al., 2023, Martinez-Finkelshtein et al., 5 Jun 2025).