Bivariate Cayley-Hamilton Theorem

Updated 14 November 2025

The bivariate Cayley-Hamilton theorem extends the classical result to two non-commuting variables, establishing a foundational bivariate characteristic identity for matrix tuples.
It underpins robust lower bound arguments in algebraic complexity, particularly impacting the analysis of algebraic branching programs and formula size constraints.
The theorem employs depth reduction and rank-based proof techniques to enforce an Ω(n²) lower bound, thereby linking invariant theory with matrix rigidity concepts.

The Bivariate Cayley-Hamilton theorem is a generalization of the classical Cayley-Hamilton theorem to the setting of two non-commuting variables interacting via polynomial identities on matrix tuples. It provides fundamental structural results that underlie a variety of lower bound arguments in algebraic complexity theory, particularly in the analysis of algebraic branching programs (ABPs), formula size, and related non-commutative models. The theorem serves as an algebraic constraint that bridges aspects of invariant theory, structural rank arguments, and dimension-counting in polynomial spaces, and is especially prominent in recent proofs of strong lower bounds for ABPs and formulas.

1. Context and Statement of the Bivariate Cayley-Hamilton Theorem

The classical Cayley-Hamilton theorem states that every square matrix $A$ over a commutative ring satisfies its own characteristic polynomial:

$p_A(A) = 0,\quad p_A(\lambda) = \det(\lambda I - A).$

In the context of tensor, non-commutative, or algebraic branching program complexity, one considers polynomials in two variables $X, Y$ , where $YX \neq XY$ , and investigates the ideals of polynomial identities they satisfy under simultaneous evaluation at matrix tuples. The bivariate Cayley-Hamilton theorem asserts that for any $n \times n$ matrices $X, Y$ , there exists a nontrivial bivariate polynomial with matrix coefficients, vanishing when $X, Y$ are specialized as such matrices.

A canonical formalization is: Let $X, Y$ be non-commuting variables, and let $\mathbb{F}\langle X, Y\rangle$ denote the free associative algebra. Over any field $\mathbb{F}$ , for any $n$ , there exists a nonzero bivariate non-commutative polynomial $P(X, Y)$ of degree $O(n)$ such that for all $X, Y \in \mathrm{Mat}_n(\mathbb{F})$ ,

$P(X, Y) = 0.$

The minimal such polynomial is the matrix bivariate characteristic identity.

The algebraic ramifications of such non-commutative identities provide structural constraints that power a series of dimension and rank arguments central to algebraic complexity lower bounds.

2. Role in ABP and Formula Lower Bounds

In algebraic complexity, especially in the analysis of ABPs, the bivariate Cayley-Hamilton theorem is a key tool for handling complexity of homogeneous polynomials evaluated at matrix tuples. The work of Chatterjee, Kumar, She, and Volk leverages such identities to perform depth reduction on ABPs and to show that any ABP of size significantly less than $n^2$ cannot compute certain high-degree polynomials, such as the "sum of $n$ th powers" $f_n(x) = \sum_{i=1}^n x_i^n$ (Chatterjee et al., 2019).

Concretely, when a layered ABP computes $f_n$ through linear or low-degree transitions, the global computation can be associated with a bivariate polynomial in the spaces spanned by $X$ and $Y$ (coming from variables or edge labels in adjacent layers). The vanishing of certain bivariate commutators or high-degree monomials imposes unavoidable rank constraints via the bivariate Cayley-Hamilton theorem.

This structural constraint underlies robust lower bounds: any attempt to keep the ABP shallow or small while still computing $f_n$ is defeated, since the bivariate Cayley-Hamilton theorem forces a minimal width or number of states at each layer to avoid trivializing the computation due to unavoidable dependencies among matrix monomials.

3. Proof Techniques: Error Accumulation and Homogeneous Lower Bounds

The main technical route involves a two-phase argument:

Depth reduction (error accumulation): Arbitrary ABPs are iteratively compressed by slicing away the central layers and substituting their action by expressions in bivariate polynomials (with structured "error" terms). At each step, the introduced error term takes the form of bilinear expressions $\sum_j P_j(x) Q_j(x)$ where $P_j$ and $Q_j$ are lower-degree (and often nontrivial) polynomials.
Homogeneous lower bound robustness: Using the bivariate Cayley-Hamilton theorem, it is shown that the minimal number of terms needed to decompose such polynomials in the ABP's variables—subject to the identities any matrix tuple must satisfy—remains quadratic in $n$ , modulo low-rank error terms. Thus, even after multiple error-accumulating depth reductions, the ABP must maintain size at least $\Omega(n^2)$ .

The theorem's geometric manifestation arises in bounding the dimension of the variety of matrices or the degree of subvarieties cut out by these bivariate polynomials. This is analogous to Noether-Lefschetz-type constraints on projective hypersurfaces, as is alternatively exploited in degree-restricted strength decompositions (Gesmundo et al., 2022).

4. Applications Beyond Classical ABPs

The bivariate Cayley-Hamilton framework generalizes to non-commutative and set-multilinear models:

Non-commutative ABPs: When evaluating non-commuting variables $X, Y$ as operators or as distinct variable layers, the bivariate Cayley-Hamilton theorem bounds the minimal size or width required for ABPs to represent certain classes of polynomials. For example, in the analysis of interval-multilinear and set-multilinear ABPs, similar polynomial identities restrict the representational capacity, leading to exponential-size lower bounds for explicit polynomials such as the permanent (Arvind et al., 2015).
Matrix rigidity and depth reduction: The analogy between Cayley-Hamilton-like identities and matrix rigidity is explicit in how depth-reduction lemmas trade off ABP layers for structured low-degree errors. This mechanism relies on the impossibility of representing all monomials independently due to the inherent relations among matrix powers and commutators encoded in the bivariate Cayley-Hamilton theorem (Chatterjee et al., 2019).
Robustness to error: The theorem ensures that small errors—in the form of structured bilinear or lower-degree terms—do not suffice to bypass the rank or dimension bottleneck, preserving the $\Omega(n^2)$ lower bound under perturbation.

5. Comparison and Consequences in Complexity Theory

The bivariate Cayley-Hamilton theorem is central to separating the computational power of various algebraic models:

Model	Lower Bound via Bivariate Cayley-Hamilton	Remark
ABP (unrestricted)	$\Omega(n^2)$ for $f_n$	Robust for low-degree labels
ABP (homogeneous)	$\Omega(n^2)$ for $f_n$	Inherited from [K19]
Formula	$\Omega(n^2)$ for $e_{0.1n}(x)$	Matches upper bound
Depth-3 formula	$\Omega(n^2)$	Ben-Or's construction optimal

The strong lower bound for ABPs arises only for explicit polynomials exhibiting maximal independence modulo bivariate identities. In contrast, polynomials such as the determinant or permanent, while requiring exponential size in highly restricted ordered or monotone models, admit polynomial-size ABPs in general.

Among the main consequences:

Any improvement in ABP lower bounds for explicit polynomials—relying on the bivariate Cayley-Hamilton structural barrier—directly impacts longstanding open problems such as the determinantal complexity of explicit polynomials, due to provable polynomial equivalence between ABP and determinantal models (Chatterjee et al., 2023).
The error-accumulation plus robust homogeneous lower bound paradigm, of which the bivariate Cayley-Hamilton theorem is the algebraic engine, forms the core of current complexity lower bound techniques, with the potential to generalize to higher-arity or multi-matrix versions and to yield super-quadratic bounds for more general classes of polynomials.

6. Open Problems and Future Directions

The full expressive power and limitations of bivariate Cayley-Hamilton-type identities in algebraic complexity are under active investigation:

Higher-arity extensions: What is the precise complexity-theoretic impact of multivariate (more than two variable) generalizations, both in commutative and non-commutative settings?
Tightness and explicitness: Identifying explicit polynomials (beyond power-sum and Shioda families) for which the bivariate Cayley-Hamilton theorem's obstruction is tight remains a central challenge (Gesmundo et al., 2022).
Orbit-closure and algebraic geometry: Understanding the equations and defining ideals associated with matrix tuple orbit-closures (as encoded by such theorems) is crucial to geometric complexity theory and the paper of border rank phenomena.
Derandomization and identity testing: The underlying identities are also relevant for the construction of hitting sets and generators in identity testing problems for ABPs with bounded-width, ordered, or monotone restrictions (Jansen et al., 2010). The corresponding varieties' dimension and rank structure inform derandomization efforts.

The bivariate Cayley-Hamilton theorem thus constitutes an algebraic barrier in polynomial computation models, serving both as a source of robust lower bounds and as a beacon for future research in algebraic and geometric complexity.