pith. sign in

arxiv: 1907.10677 · v1 · pith:NUR5G5BUnew · submitted 2019-07-23 · 💻 cs.SC · cs.NA· math.NA

Upper Hessenberg and Toeplitz Bohemians

Eunice Y. S. Chan , Robert M. Corless , Laureano Gonzalez-Vega , J. Rafael Sendra , Juana Sendra This is my paper

Pith reviewed 2026-05-24 17:06 UTC · model grok-4.3

classification 💻 cs.SC cs.NAmath.NA
keywords Bohemian matricesupper Hessenberg matricescharacteristic polynomialsmatrix heightnormal matricesstable matricesToeplitz matrices
0
0 comments X

The pith

Upper Hessenberg Bohemian matrices with entries from a fixed small set have characteristic polynomials whose maximal height grows exponentially with matrix order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines Bohemian matrices whose entries come from a small fixed population such as {-1, 0, +1}. Attention is restricted to the upper Hessenberg case, and then further to those matrices whose characteristic polynomials attain maximal height. This restriction permits explicit identification of the polynomials, derivation of bounds on their height, and an accurate asymptotic conjecture. A central result states that the lower bound on this maximal height is exponential in the matrix order n, while the matrices themselves retain constant height. Theorems are also supplied that count the normal matrices and the stable matrices inside these families.

Core claim

For upper Hessenberg Bohemian matrices over populations such as {-1, 0, +1}, the characteristic polynomials of maximal height can be identified explicitly; their height admits useful bounds whose lower bound is exponential in the matrix order, together with an asymptotic conjecture for that height; the families also contain explicitly countable numbers of normal matrices and of stable matrices.

What carries the argument

Upper Hessenberg Bohemian matrices whose characteristic polynomials achieve maximal height; the restriction to this subclass allows explicit identification of the polynomials and derivation of height bounds.

If this is right

  • The height of the characteristic polynomial grows exponentially with matrix order while entry size stays fixed.
  • Explicit closed forms exist for the polynomials that achieve the maximal height.
  • The number of normal matrices inside each family is finite and can be counted exactly.
  • The number of stable matrices inside each family is finite and can be counted exactly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same exponential growth in characteristic height may appear in other structured Bohemian families such as Toeplitz matrices.
  • Characteristic height could serve as a practical filter for locating matrices with extreme spectral properties even when the entry set is bounded.
  • The explicit polynomials identified here may satisfy simple recurrences that could be used to generate larger examples without enumeration.

Load-bearing premise

Restricting attention to the subclass of upper Hessenberg Bohemian matrices whose characteristic polynomials have maximal height permits explicit identification of those polynomials together with useful bounds.

What would settle it

A direct computation or proof for some order n that produces an upper Hessenberg Bohemian matrix whose characteristic polynomial height is strictly smaller than the claimed exponential lower bound.

Figures

Figures reproduced from arXiv: 1907.10677 by Eunice Y. S. Chan, J. Rafael Sendra, Juana Sendra, Laureano Gonzalez-Vega, Robert M. Corless.

Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

We look at Bohemians, specifically those with population $\{-1, 0, {+1}\}$ and sometimes $\{0,1,i,-1,-i\}$. More, we specialize the matrices to be upper Hessenberg Bohemian. From there, focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the numbers of normal matrices and the numbers of stable matrices in these families.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper examines upper Hessenberg Bohemian matrices with entries drawn from {-1,0,1} or the five-element set {0,1,i,-1,-i}. Restricting attention to those matrices whose characteristic polynomials achieve maximal height permits explicit identification of the polynomials, derivation of bounds on height, and an asymptotic conjecture. The central results include an exponential lower bound on maximal characteristic height (while matrix height remains 1) together with theorems counting the normal matrices and the stable matrices within each family.

Significance. If the explicit identifications and bounds are correct, the work supplies concrete constructions showing that coefficient height in characteristic polynomials can grow exponentially even when all matrix entries are drawn from a fixed finite set of height 1. The counting theorems for normal and stable matrices furnish quantitative information that may be useful in symbolic computation and stability studies of structured matrices. The restriction to maximal-height cases is presented as the device that yields closed forms; this is a methodological strength when the resulting objects are fully enumerated and verified.

major comments (2)
  1. [Abstract] The exponential lower bound on maximal characteristic height is the load-bearing claim, yet the manuscript provides no derivation details, error analysis, or verification data for the enumerated cases that achieve the bound (Abstract).
  2. [Abstract] The post-hoc selection of only maximal-height matrices is used to enable explicit identification, but the paper does not address whether the resulting polynomials are representative or whether the exponential growth persists without this selection (Abstract and theorems on normal/stable matrices).
minor comments (2)
  1. [Abstract] The abstract asserts 'explicit identification' and 'useful bounds' without indicating the matrix orders for which the identification is complete or the precise definition of 'height' employed.
  2. Notation for the two entry sets should be introduced once and used consistently when stating the families for the normal-matrix and stable-matrix counts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report. The two major comments both concern the presentation and scope of the central claims in the abstract; we respond point by point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] The exponential lower bound on maximal characteristic height is the load-bearing claim, yet the manuscript provides no derivation details, error analysis, or verification data for the enumerated cases that achieve the bound (Abstract).

    Authors: The abstract states the bound concisely, as is conventional. The body of the manuscript supplies the explicit polynomials, the recurrence used to derive the exponential lower bound, and the enumeration for small orders. We will revise the abstract to include a brief pointer to the relevant sections and add a short verification table (or appendix) listing the maximal-height polynomials and their heights for n up to 12, together with the computational method used to confirm them. revision: yes

  2. Referee: [Abstract] The post-hoc selection of only maximal-height matrices is used to enable explicit identification, but the paper does not address whether the resulting polynomials are representative or whether the exponential growth persists without this selection (Abstract and theorems on normal/stable matrices).

    Authors: The manuscript explicitly frames the restriction to maximal-height matrices as the device that permits closed-form identification; it does not claim these polynomials are representative of the full set of Bohemian matrices. The exponential lower bound and the counting theorems for normal and stable matrices are stated only for the maximal-height subfamily. We will insert a clarifying sentence in the introduction and abstract noting the deliberate scope and that growth rates outside the maximal-height case lie outside the present work. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results rest on explicit symbolic identification and enumeration of upper Hessenberg Bohemian matrices (entries from {-1,0,1} or the five-element set) whose characteristic polynomials achieve maximal height. This identification directly supplies the exponential lower bound on height (while matrix height remains 1) and the counting theorems for normal and stable matrices. No step reduces a claimed prediction or bound to a fitted parameter, self-referential definition, or load-bearing self-citation; the derivation chain is self-contained against external enumeration and symbolic computation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper rests on standard definitions of Bohemian matrices and Hessenberg structure with no new free parameters or invented entities stated in the abstract.

axioms (2)
  • domain assumption Bohemian matrices are those whose entries belong to a small discrete population such as {-1,0,+1}.
    Foundational definition invoked throughout the abstract.
  • domain assumption Upper Hessenberg matrices have zeros below the first subdiagonal.
    Standard structural restriction used to specialize the family.

pith-pipeline@v0.9.0 · 5657 in / 1276 out tokens · 23397 ms · 2026-05-24T17:06:30.888241+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Constants phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    The lower bound for the maximal characteristic height is exponential in the order of the matrix; ... τ_n between F_{2n+1}/(n+1) and F_{2n+1}

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    Baez, The beauty of roots, Available at: https://johncarlosbaez.wordpress.com/2011/12/11/the- beauty-of-roots/, (2011)

    J. Baez, The beauty of roots, Available at: https://johncarlosbaez.wordpress.com/2011/12/11/the- beauty-of-roots/, (2011)

    work page 2011

  2. [2]

    Beaucoup, P

    F. Beaucoup, P. Borwein, D. W. Boyd, and C. Pinner , Multiple roots of [−1, 1] power series, Journal of the London Mathematical Society, 57 (1998), pp. 135–147

    work page 1998

  3. [3]

    Borwein , Computational excursions in analysis and number theory , Springer Science & Business Media, 2012

    P. Borwein , Computational excursions in analysis and number theory , Springer Science & Business Media, 2012

    work page 2012

  4. [4]

    Borwein and L

    P. Borwein and L. J ¨orgenson, Visible structures in number theory , The American Mathe- matical Monthly, 108 (2001), pp. 897–910

    work page 2001

  5. [5]

    Borwein and C

    P. Borwein and C. Pinner , Polynomials with {0,+1,-1} coefficients and a root close to a given point, Canadian Journal of Mathematics, 49 (1997), pp. 887–915

    work page 1997

  6. [6]

    Briat, Sign properties of Metzler matrices with applications , Linear Algebra and its Appli- cations, 515 (2017), pp

    C. Briat, Sign properties of Metzler matrices with applications , Linear Algebra and its Appli- cations, 515 (2017), pp. 53–86

    work page 2017

  7. [7]

    N. D. Cahill, J. R. D’Errico, D. A. Narayan, and J. Y. Narayan , Fibonacci determinants, The College Mathematics Journal, 33 (2002), pp. 221–225

    work page 2002

  8. [8]

    E. Y. S. Chan , A comparison of solution methods for Mandelbrot-like polynomials , Electronic Thesis and Dissertation Repository, (2016). https://ir.lib.uwo.ca/etd/4028

    work page 2016

  9. [9]

    E. Y. S. Chan and R. M. Corless , A new kind of companion matrix , Electronic Journal of Linear Algebra, 32 (2017), pp. 335–342

    work page 2017

  10. [10]

    E. Y. S. Chan and R. M. Corless , Minimal height companion matrices for Euclid polynomials, Mathematics in Computer Science, (2018), https://doi.org/10.1007/s11786-018-0364-2

    work page doi:10.1007/s11786-018-0364-2 2018

  11. [11]

    E. Y. S. Chan, R. M. Corless, L. Gonzalez-Vega, J. R. Sendra, and J. Sendra , Algebraic linearizations of matrix polynomials , Linear Algebra and its Applications, 563 (2019), pp. 373–399

    work page 2019

  12. [12]

    R. M. Corless , Generalized companion matrices in the Lagrange basis, in Proceedings EACA, Santander, Spain: Universidad de Cantabria, 2004, pp. 317–322

    work page 2004

  13. [13]

    R. M. Corless and P. W. Lawrence , Mandelbrot polynomials and matrices. In preparation

    work page

  14. [14]

    R. M. Corless and P. W. Lawrence , The largest roots of the Mandelbrot polynomials , in Computational and Analytical Mathematics, Springer, 2013, pp. 305–324

    work page 2013

  15. [15]

    R. M. Corless and S. Thornton , Visualizing eigenvalues of random matrices , ACM Com- munications in Computer Algebra, 50 (2016), pp. 35–39, https://doi.org/10.1145/2930964. 2930969

    work page doi:10.1145/2930964 2016

  16. [16]

    R. M. Corless and S. E. Thornton , The Bohemian eigenvalue project , ACM Communica- tions in Computer Algebra, 50 (2016), pp. 158–160

    work page 2016

  17. [17]

    L. M. DeAlba , Determinants and eigenvalues , in Handbook of Linear Algebra, L. Hogben, ed., Chapman and Hall/CRC, 2013, ch. 4

    work page 2013

  18. [18]

    Embree , Pseudospectra, in Handbook of Linear Algebra, L

    M. Embree , Pseudospectra, in Handbook of Linear Algebra, L. Hogben, ed., Chapman and Hall/CRC, 2013, ch. 23

    work page 2013

  19. [19]

    F asi and G

    M. F asi and G. M. N. Porzio , Determinants of normalized Bohemian upper Hessenberg matrices. May 2019, http://eprints.maths.manchester.ac.uk/2709/. 24 E. Y. S. CHAN, ET AL

    work page 2019

  20. [20]

    Gear, A simple set of test matrices for eigenvalue programs , Mathematics of Computation, 23 (1969), pp

    C. Gear, A simple set of test matrices for eigenvalue programs , Mathematics of Computation, 23 (1969), pp. 119–125

    work page 1969

  21. [21]

    F. J. Hall and Z. Li , Sign pattern matrices, in Handbook of Linear Algebra, L. Hogben, ed., Chapman and Hall/CRC, 2013, ch. 42

    work page 2013

  22. [22]

    Hardin and G

    D. Hardin and G. Strang , A thousand points of light , The College Mathematics Jour- nal, 21 (1990), pp. 406–409, https://doi.org/10.1080/07468342.1990.11973345, https:// doi.org/10.1080/07468342.1990.11973345, https://arxiv.org/abs/https://doi.org/10.1080/ 07468342.1990.11973345

    work page doi:10.1080/07468342.1990.11973345 1990

  23. [23]

    Higham , Bohemian matrices in numerical linear algebra

    N. Higham , Bohemian matrices in numerical linear algebra . Available at http://www.maths. manchester.ac.uk/∼higham/conferences/bohemian/higham bohemian18.pdf(June 20, 2018)

    work page 2018

  24. [24]

    Janjic , Words and linear recurrences, Journal of Integer Sequences, 21 (2018), p

    M. Janjic , Words and linear recurrences, Journal of Integer Sequences, 21 (2018), p. 3

    work page 2018

  25. [25]

    Jeffries, V

    C. Jeffries, V. Klee, and P. V an den Driessche , When is a matrix sign stable? , Canadian Journal of Mathematics, 29 (1977), pp. 315–326

    work page 1977

  26. [26]

    Kilic and D

    E. Kilic and D. Tasci , On the generalized Fibonacci and Pell sequences by Hessenberg ma- trices, Ars Combin, 94 (2010), pp. 161–174

    work page 2010

  27. [27]

    M. C. Lettington , Fleck’s congruence, associated magic squares and a zeta identity , Func- tiones et Approximatio Commentarii Mathematici, 45 (2011), pp. 165–205

    work page 2011

  28. [28]

    A. M. Odlyzko , Zeros of polynomials with 0,1 coefficients , in Algorithms Seminar, B. Salvy, ed., no. 2130, December 1993, pp. 169–172, http://citeseerx.ist.psu.edu/viewdoc/ download?doi=10.1.1.47.9327&rep=rep1&type=pdf#page=175

    work page 1993

  29. [29]

    Polya, How to solve it: A new aspect of mathematical method , Princeton University Press, 2014

    G. Polya, How to solve it: A new aspect of mathematical method , Princeton University Press, 2014

    work page 2014

  30. [30]

    Shattuck , Combinatorial proofs of determinant formulas for the Fibonacci and Lucas polynomials, The Fibonacci Quarterly, 51 (2013), pp

    M. Shattuck , Combinatorial proofs of determinant formulas for the Fibonacci and Lucas polynomials, The Fibonacci Quarterly, 51 (2013), pp. 63–71

    work page 2013

  31. [31]

    N. J. A. Sloane , The on-line encyclopedia of integer sequences (a062110) . Published elec- tronically at https://oeis.org/A062110 (June 24, 2018)

    work page 2018

  32. [32]

    N. J. A. Sloane , The on-line encyclopedia of integer sequences (a105306) . Published elec- tronically at https://oeis.org/A105306 (June 20, 2018)

    work page 2018

  33. [33]

    Tao and V

    T. Tao and V. Vu , Random matrices have simple spectrum, Combinatorica, 37 (2017), pp. 539– 553

    work page 2017

  34. [34]

    Taussky, Matrices of rational integers , Bulletin of the American Mathematical Society, 66 (1960), pp

    O. Taussky, Matrices of rational integers , Bulletin of the American Mathematical Society, 66 (1960), pp. 327–345

    work page 1960

  35. [35]

    Taussky , Some computational problems involving integral matrices , JOURNAL OF RESEARCH OF THE NATIONAL BUREAU OF STANDARDS SECTION B- MATHEMATICAL SCIENCES, 65 (1961), pp

    O. Taussky , Some computational problems involving integral matrices , JOURNAL OF RESEARCH OF THE NATIONAL BUREAU OF STANDARDS SECTION B- MATHEMATICAL SCIENCES, 65 (1961), pp. 15–17

    work page 1961

  36. [36]

    S. E. Thornton , The characteristic polynomial database . Available at http:// bohemianmatrices.com/cpdb (Sept. 7, 2018)

    work page 2018

  37. [37]

    Available at https://en.wikipedia.org/wiki/ Composition (combinatorics) (May 15, 2019)

    Wikipedia, Composition (combinatorics) . Available at https://en.wikipedia.org/wiki/ Composition (combinatorics) (May 15, 2019)

    work page 2019