bibkey: garciamarco2025toeplitzfactors
authors: Ignacio García-Marco, Irene Márquez-Corbella, Daniel Seco
year: 2025
title: "On the minimum number of Toeplitz factors of a matrix"
doi: 10.48550/arXiv.2506.16432
url: https://arxiv.org/abs/2506.16432v1
claim: "Section 5.2, Conjecture 5.1 proposes a positive dimension-independent number C such that every complex square matrix is a product of an invertible left matrix and C ordinary Toeplitz matrices; its subsequent numerical consequence is a separate assertion."
strata_touched:
  - D5/S3/Quantum/Matrix/Toeplitz/FactorizationRefutation
license: citation-only
triage: anchor

Verified locator

DOI: 10.48550/arXiv.2506.16432

Source: https://arxiv.org/abs/2506.16432v1

On the minimum number of Toeplitz factors of a matrix

The source is arXiv:2506.16432v1, submitted 2025-06-19. The original source archive contains Toeplitz-GMS.tex. The abstract/history, HTML and original TeX give the same conjecture. The arXiv history inspected on 2026-10-28 lists only v1 and no journal reference. The TeX uses \date{\today}; a rendered creation date does not identify another version.

The introduction, equation (1), defines ordinary complex Toeplitz matrices by

M_{i,j}=M_{i+1,j+1} for all i,j=1,…,n-1.

The introduction defines a Toeplitz decomposition as an ordered tuple of same-size factors whose product is the matrix. It defines Toep(M) as the minimum such length and Toep_n as the maximum over all complex n by n matrices. Section 5.2 defines Toep_n^Inv as the minimum number of factors sufficient for every invertible matrix of that size.

Conjecture 5.1, Section 5.2, original TeX lines 1170–1175 (super-conjecture):

There exists a constant C > 0 such that every matrix can be expressed as a product of an invertible matrix and C Toeplitz matrices. In particular, Toep_n ≤ Toep_n^Inv + C.

The ambient field is the complex numbers, and singular matrices are included. The rank-(n−1) proposition in Section 4 explicitly writes M = P T_1 T_2, with P invertible, and its proof writes M = P_0 rref(M). This fixes the left-multiplier interpretation. The positive integer C counts factors and is uniform in both dimension and matrix. Identity padding makes exact and upper-bounded lengths equivalent. Two independently free invertible endpoint multipliers constitute a different statement.

The Lean claim is the first factorization assertion F in full. Its refutation does not logically refute the numerical assertion J introduced by “In particular”. No converse J ⇒ F or unbounded numerical gap is established. Conjecture 5.2 concerns row equivalence to a quasi-Toeplitz matrix and is excluded from this target; Conjecture 5.3 concerns exact worst-case Toeplitz numbers and is also separate.

Parameter obstruction and sources

Parameter-count obstructions for structured matrix products are established methodology. Ye and Lim, Every Matrix is a Product of Toeplitz Matrices, Found. Comput. Math. 16 (2016), 577–598, use dimension counts for generic lower bounds; the present source recalls this in its introduction and Section 5.3. Their general factorization of arbitrary matrices uses two endpoint multipliers, and does not give the uniformly bounded single-left-multiplier assertion F. The common-field argument uses the standard transcendence-degree bound for finitely generated fields, as in the Stacks Project, tags 030D and 030H. No novelty of this methodology is asserted.

Shivam Patel’s community submission, A dimension obstruction in every sufficiently large dimension, on MathDB’s Conjecture 5.2 page, uses the related parameter obstruction on Grassmannians for quasi-Toeplitz row spaces. Its final sentence explicitly excludes the distinct factorization conjectures. The page labels the submission as not independently checked. Its claimed binary counterexample and modular census are not premises of this proof. Avoiding a single quasi-Toeplitz row space does not by itself exclude products of an arbitrary positive number C of ordinary Toeplitz matrices; the present claim requires the common field of all C factors and a dimension chosen with C.

For a proposed positive C, choose r = 4C, n = 2r, and an r by r complex matrix X with complex entries algebraically independent over the rationals. For M = [[I,X],[0,0]], invertible left multiplication preserves its actual graph row space. A product of C Toeplitz matrices has entries in the one field generated by all C(2n−1) diagonal coefficients. A nonsingular selection of actual projected rows recovers X over that field. Its transcendence degree must therefore be both at least r² and at most C(2n−1), whereas C(2n−1)+C = r². The coefficients of the free invertible multiplier are not counted.

Bounded source and prior-art scope

The source comparison includes the introduction, all rank constructions, all of Section 5.2, the neighboring questions and references. DataCite metadata identifies version 1. No journal successor was identified in the supplied Crossref/DataCite/OpenAlex title and author checks. The refreshed arXiv title/topic search inspected the returned 79 records, including the original source, Ye–Lim and the later recurrence-factorization lead arXiv:2609.27268; no matching resolution was identified in that searched scope. This is not exhaustive literature clearance or a priority claim.

The MathDB reading on 2026-10-28 returned HTTP 200 for page 368924, whose structured answer count is zero and whose visible text says no solutions have been posted. Page 368925 has one submission for the separate Conjecture 5.2. These are page readings, not verification of global openness. Indexed repository issue and pull-request searches include all states and distinguish exact target #23130 from unrelated coefficient-Toeplitz, Hurwitz and factorization questions. Index latency and unindexed results remain outside those readings.