Algebraic approach to the inverse spectral problem for rational matrices

Abstract: We consider the problem of reconstruction of an $n\times n$ matrix with coefficients depending rationally on $x\in \mathbb P^1$ from the data of: (a) its characteristic polynomial and (b) a line bundle of degree $g+n-1$, with $g$ the geometric genus of the spectral curve, represented by a choice of $g+n+1$ points forming a (non-positive) divisor of the given degree. We thus provide a reconstruction formula that does not involve transcendental functions; this includes formulas for the spectral projectors and for the change of line bundle, thus integrating the isospectral flows. The formula is a single residue formula which depends rationally on the coordinates of the points involved, the coefficients of the spectral curve, and the position of the finite poles of $L$. We also discuss the canonical bi-differential associated with the Lax matrix and its relationship with other bi-differentials that appear in Topological Recursion and integrable systems.

• The paper presents an explicit algebraic reconstruction formula based solely on residue calculus to recover rational Lax matrices from given spectral data.
• It replaces transcendental techniques with purely rational computations, offering a more computationally efficient approach for integrable systems.
• The method exhibits strong practical implications by supporting numerical and symbolic evaluations, preserving rationality for applications in moduli theory and arithmetic geometry.

Algebraic Construction for the Inverse Spectral Problem of Rational Matrices

Problem Setting and Motivation

The paper addresses the algebraic reconstruction of an $n \times n$ rational matrix $L(x)$ from its spectral data: specifically, given a characteristic equation (the spectral curve $E(x, y) = 0$ with $\deg_y E = n$) and a line bundle of degree $g+n-1$ (with $g$ the geometric genus of the spectral curve), the problem is to recover $L(x)$. This is the classical non-abelian inverse spectral problem, central in the integrable systems context and related to the abelianization of Hitchin systems.

Historically, solutions to this problem hinged on transcendental data: explicit formulas required Riemann theta functions, Abel maps, and period computations on Riemann surfaces, rendering these approaches impractical for generic data, especially when computational effectiveness or number-theoretic structures are of interest. The paper's main claim is the derivation of a fully algebraic, residue-based reconstruction formula, bypassing transcendental intermediaries, relying only on rational computations parameterized by explicitly given geometric data.

Algebraic Reconstruction Framework

The input is:

• The explicit form of $E(x, y)$, which defines the spectral curve $\mathcal{C}$. The geometric data consists of the Newton polygon, its interior to determine the genus $g$, and polynomials $a_j(x)$ defining $E$.
• A normalization point $z_0 \in \mathbb{P}^1$ (not a branch point of $X:\mathcal{C}\to\mathbb{P}^1$), together with pre-images $\{z_0^{(\ell)}\}$.
• A divisor $\mathscr{D}$ of degree $g$ (none of the points special).
• An additional point $p_0$, so that the line bundle’s divisor is $\mathscr{D} - p_0 + \sum_\ell z_0^{(\ell)}$.

The main algebraic object is a Cauchy kernel $_{\mathscr{D}-p_0}(p,q)$, a differential of the third kind, characterized by poles at $p=q$ and $p=p_0$ (residues $\pm 1$), and zeros at $\mathscr{D}$. This explicit kernel, constructed via interpolation constraints enforced through a rational system, underlies the construction of the matrix solution.

Main theorem: The Lax matrix $L(x)$ is given by a residue formula (Theorem 1), where $L_{ab}(x)$ is a sum of residues in $y$ coordinated entirely by the rational geometry of $E$, the normalization data, and the positions of the divisor points. The construction does not invoke transcendental precomputations—if the geometric data are rational, $L(x)$ is rationally computable.

Explicit Residue and Kernel Construction

A detailed analysis leads to an explicit formula for the Cauchy kernel (Proposition 1). For $E(x, y)$ with suitable genericity conditions on leading coefficients to ensure uniqueness of poles and regularity at infinity (cf. the "demonic" assumption on Newton polygons), the construction utilizes the unique differential

$$_{\mathscr{D}-p_0}(p,q) = \frac{Q((x,y); (z,w)) x}{E_y(x,y)}$$

where $Q$ is a rational function determined via the imposition of vanishing at the divisor points $\mathscr{D}$. The construction uses only rational operations (solving a linear system defined by the interpolation constraints), so for explicit input, all data remains algebraic/rational.

The explicit $L(x)$ recovers the characteristic equation: the constructed matrix has the prescribed spectral curve, and the eigenvector data is encoded in the algebraically defined line bundle.

Relations to Projectors and Correlators

The paper provides explicit, algebraic formulas for spectral projectors $\Pi(x, y)$ (via adjugacy and normalized traces), as well as the associated spectral bi-differential $B(p, q)$, central tools for tau-function computations and topological recursion. These formulas, previously known only in theta function or transcendental form, are herein given by rational residue evaluations dependent solely on the input data.

The multi-point correlators $W_N$ of the associated multi-KP hierarchy are similarly expressed as symmetrized residue products over the Cauchy kernel, providing an algebraic path to explicitly compute quantities intimately connected to intersection theory and tau-functions.

Comparison to Other Methods and Theoretical Implications

A critical aspect of the paper is the systematic exposition of how the new algebraic residue formulas replace the transcendental machinery (Szegő kernels, Abel maps, Riemann theta functions, period matrices). The only transcendental structure invoked is for conceptual comparison: the algebraic Cauchy kernel’s correlation with the Szegő kernel and fundamental bi-differential is made explicit (Section 6).

Notably, the approach allows for explicit computation over $\mathbb{Q}$—if all data are rational, so is the output. This is in stark contrast to the theta function machinery, which generically only operates over $\mathbb{C}$. The construction is, in principle, implementable on any computer algebra system, handling arbitrary genus and matrix size, with the only potential obstruction coming from the solution of a finite, explicitly given system of polynomial equations (the generic case being regular and computationally tractable).

Strong and Contradictory Claims

• Explicit, purely algebraic reconstruction formula: The proposed formula allows for rational computation of $L(x)$ without the intervention of transcendental functions, in contradiction to the widespread reliance on classical (theta function, Abel map) approaches.
• No transcendental integration required: All ingredients are rational or algebraic; the method circumvents any need for period integrals or basis selection in homology, notoriously problematic in previous methods.
• Compatibility with number-theoretic structures: The construction preserves rationality precisely—if all input data is rational, so is the output, which is not possible in the theta functional framework.

Numerical and Symbolic Examples

The paper demonstrates several explicit examples, including nontrivial genus-2, $n = 3$ Lax matrices, with all coefficients and divisor data entirely rational. The examples are given in explicit polynomial and rational form, supporting the practicability and computational accessibility of the approach.

Implications and Future Directions

The algebraic residue methodology fundamentally alters the computational landscape of the non-abelian inverse spectral problem. The implications are:

• Efficient Numerical Computation: The method gives a clear algorithm for symbolic or numeric computation of rational Lax matrices, spectral projectors, and correlators associated to arbitrary spectral data, compatible with modern CAS and effective for explicit examples.
• Abelianization and Moduli Theory: The explicit algebraic correspondence between the space of rational matrices and abelian data (spectral curves and line bundles) accords with the Hitchin fibration, making the approach relevant for the study of integrable systems, moduli of Higgs bundles, and Hamiltonian flows on character varieties.
• Topological Recursion and Quantum Geometry: By producing rational analogs of the bidifferential and correlator objects central to the Chekhov-Eynard-Orantin topological recursion, the methodology facilitates explicit algebraic computations of tau-functions and intersection numbers in Gromov-Witten theory.
• Arithmetic Applications: The method invites further investigation into the arithmetic of spectral curves and Lax representations, perhaps linking the explicit moduli-theoretic description to problems in arithmetic geometry.

Potential future developments may include extension to more general monodromy or Higgs data (e.g., including parabolic structures, irregular singularities), extension to difference (discrete) Lax operators, and the development of efficient software packages implementing the method for explicit computation in high genus and high rank.

Conclusion

This paper establishes a structurally transparent, explicit, and algebraic framework for the inverse spectral problem for rational Lax matrices. The rational residue formula, together with the explicit Cauchy kernel, fully replaces transcendental methods by algebraic manipulations. This approach clarifies the geometric mechanism underlying the non-abelianization/abelianization correspondence, enables fully explicit computation in applications to integrable systems, moduli theory, and algebraic geometry, and is compatible with symbolic and arithmetic computation frameworks.

The result provides a new standard for computability in the theory of isospectral deformations, integrable hierarchies, and algebraic approaches to matrix spectral problems.