A new mathematical preprint has identified the exact leading constant in Schäffer’s determinant–inverse matrix inequality: after the optimal universal constant S_n is divided by the square root of the dimension n, the result tends to √e as n becomes large. The paper also gives an explicit family of norms and invertible matrices whose normalized ratio approaches the same value.
The paper presents this as the precise first-order, dimension-dependent loss associated with allowing arbitrary Banach-space geometry. The central statement is a mathematical limit, not an empirical trend measured in a population or dataset.
A question about dimension and geometry
The quantity S_n is defined as the smallest universal constant that makes the determinant–inverse inequality hold for invertible operators on every n-dimensional complex Banach space. The question is not about one particular matrix: it asks how large that constant must be when the underlying space can have different Banach-space geometries.
The study is entirely theoretical. There is no empirical population, dataset or statistical sample. Instead, the objects under examination are norms and invertible operators, represented as matrices acting on C^n, with the dimension indexed by n.
The comparison is mathematical rather than experimental. There is no control group; the paper compares possible norms, spectra and operators within the framework of the inequality itself.
Turning many possible norms into a spectral problem
The proof begins by reducing the optimization over norms and invertible matrices to an optimization over spectra — the sets of eigenvalues associated with the operators. This reduction uses a representation attributed in the paper to Gluskin, Meyer and Pajor, allowing the problem to be studied through carefully chosen spectral configurations.
The authors then connect the matrix question to an interpolation problem and to norms of polynomial coefficients. A Hölder-duality argument shows that every monic polynomial Q supplies a lower bound for the prescribed-spectrum interpolation functional, with the polynomial’s coefficient norm linked to the determinant term in the original inequality.
That link turns the search for a difficult matrix example into a sequence of extremal problems. The paper studies how zeros should be distributed, how a polynomial should be chosen under a divisibility condition, and how the resulting analytic object can be realized by an explicit matrix.
Why the number √e appears
At the continuum stage of the argument, the authors optimize over possible zero densities. The sharp value of that weighted problem is e−1/2, and the only density that attains the sharp lower bound is τ(t)=log(2π/t). The paper’s asymptotic constant is tied to this logarithmic profile, which reappears in the later discrete construction.
The same value reappears in a polynomial extremal problem. For every integer L≥1, the weighted Chebyshev infimum over monic polynomials Q divisible by (z−1)^L equals e−1/2. A Chebyshev problem here means finding a monic polynomial that keeps a specified weighted size as small as possible under the stated zero condition.
The construction then discretizes the optimal logarithmic density to produce finite spectra. In the paper’s notation, this uses r_n=1−n−1/6 together with the distribution map M(u)=u(1−log u). These choices are part of the explicit recipe for the dimension-indexed examples, rather than estimates inferred from sampled data.
An explicit family reaches the bound
The analytic estimates are converted into matrix statements through a model-operator realization. For the fixed choice L=16 and a nonzero polynomial Q with the required divisibility, the largest Taylor-coefficient norm of Q B_n has the asymptotic scale C(Q)/√n: equivalently, √n‖Q B_n‖ tends to C(Q).
The resulting matrices have the prescribed distinct eigenvalues and induced norm at most 1. The model-operator identity identifies the determinant–inverse expression for that spectrum with the interpolation functional φ, so the lower bound established in the analytic part transfers directly to the matrices.
This is the key attainment result. For every dimension n≥2, the authors specify a norm and an invertible matrix, and the normalized determinant–inverse ratio for that sequence tends to √e. The construction therefore matches the upper asymptotic constant rather than merely suggesting that the constant might be attainable.
What the theorem establishes — and what remains open
Taken together, the results show that Schäffer’s upper bound is asymptotically sharp: the leading dimension-dependent loss in the general Banach-space problem is exactly captured by √e after normalization by √n. The theorem identifies the first-order behavior, while the explicit construction shows that this behavior is realized by concrete norms and matrices.
The conclusion is asymptotic, so it describes what happens as the dimension grows rather than giving a complete table of finite-dimensional values. The supplied analysis does not give a finite-n error bound for the final convergence statement, leaving the speed of approach to √e unresolved.
The construction proves the existence of one explicit near-extremal family; it does not characterize every norm or matrix that could come close to the bound. Nor does the work test empirical performance or application outcomes, because its subject is a deterministic mathematical construction involving operators, spectra and norms.
The paper’s technical route relies on spectral representation, interpolation, Hölder duality, weighted polynomial analysis and oscillatory estimates. In the appendix, the authors control the relevant phase near its critical point with stationary-phase analysis and use van der Corput estimates elsewhere.
The document is an arXiv version 1 preprint dated 20 Aug 2026. Its disclosure says that ChatGPT assisted with editing and writing, while the authors developed the mathematical results and arguments; no funding source is reported in the supplied front matter.
Paper data and sources
Original title: Schäffer's matrix inequality: the exact asymptotic constant
Authors: Samy Houache, Oleg Szehr, Rachid Zarouf
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text