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arxiv: 2401.00089 · v4 · pith:LER6FZSKnew · submitted 2023-12-29 · 🧮 math.AG · cs.SC

Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach)

Hoon Hong , Daniel Profili , J. Rafael Sendra This is my paper

Pith reviewed 2026-05-24 05:01 UTC · model grok-4.3

classification 🧮 math.AG cs.SC
keywords eigenvalue configurationsymmetric matricessymmetric polynomialsreal root countingDescartes rule of signsparametric matricesalgebraic conditionscharacteristic polynomials
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The pith

Eigenvalue configurations of two parametric real symmetric matrices reduce to real root counts of associated symmetric polynomials.

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

The paper presents an algorithm that converts the problem of determining when two parametric real symmetric matrices have a prescribed relative ordering of their eigenvalues into the task of counting real roots of certain constructed symmetric polynomials. These root counts are then evaluated using the Fundamental Theorem of Symmetric Polynomials to rewrite the counts in terms of elementary symmetric polynomials and Descartes' rule of signs to determine the number of positive roots from coefficient sign patterns. A reader would care because this yields explicit algebraic conditions on the parameters that enforce a desired eigenvalue arrangement without needing to solve the characteristic equations or compute the eigenvalues themselves. The approach applies to families of matrices where the entries are polynomials or rational functions in the parameters.

Core claim

Given two parametric real symmetric matrices and a target eigenvalue configuration, the configuration holds precisely when certain associated symmetric polynomials have a specific number of real roots; these root numbers are obtained algorithmically by expressing the polynomials via the Fundamental Theorem of Symmetric Polynomials and applying Descartes' rule of signs to count positive roots.

What carries the argument

Symmetric polynomials constructed from the input matrices whose real-root multiplicities directly encode the relative positions of the eigenvalues on the line.

If this is right

  • The algorithm outputs explicit conditions on the parameters as systems of inequalities derived from the root counts.
  • Eigenvalue interlacing or separation properties become decidable by sign-pattern analysis alone.
  • The method extends to any number of matrices by iterating the pairwise construction.
  • No numerical eigenvalue solvers are required; all steps remain algebraic.

Where Pith is reading between the lines

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

  • The same reduction might be used to certify stability regions in parametric linear systems whose matrices are symmetric.
  • Low-dimensional test cases such as 2-by-2 matrices could be checked by hand to verify the root-count predictions match direct eigenvalue calculations.
  • The approach suggests a route to semi-algebraic descriptions of eigenvalue configuration strata in the parameter space.

Load-bearing premise

The mapping from eigenvalue configurations to the root counts of the constructed symmetric polynomials is bijective and introduces no extraneous roots or missed cases for arbitrary parameter values.

What would settle it

Take two low-dimensional parametric symmetric matrices, compute their eigenvalue ordering by solving the characteristic polynomials directly, then apply the algorithm to obtain the predicted root counts; any mismatch between the observed ordering and the predicted counts for some parameter values falsifies the reduction.

read the original abstract

Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that their eigenvalues have the given configuration. We give an algorithm which expresses the eigenvalue configuration problem as a real root counting problem of certain symmetric polynomials, whose roots can be counted using the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs.

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 / 0 minor

Summary. The manuscript claims to provide an algorithm that reduces the problem of finding conditions on parameters for two given parametric real symmetric matrices to exhibit a prescribed eigenvalue configuration to the task of counting real roots of certain symmetric polynomials; these counts are then obtained via the Fundamental Theorem of Symmetric Polynomials together with Descartes' rule of signs.

Significance. If the reduction is shown to be faithful, the approach would supply an algebraic route to explicit parameter conditions for eigenvalue arrangements that avoids direct eigenvalue computation. Such a tool could be useful in parametric stability or control problems, but the absence of any worked example or verification that the root counts recover the configuration leaves the practical significance unclear.

major comments (2)
  1. [Abstract] Abstract: the reduction of the eigenvalue-configuration problem to real-root counting of the constructed symmetric polynomials is asserted without derivation steps or verification that the counts uniquely encode the desired configuration for generic parametric matrices.
  2. [Abstract] Abstract: Descartes' rule of signs is invoked to obtain the root counts, yet the rule supplies only an upper bound (number of sign changes or less by an even integer) on the number of positive real roots. The manuscript does not address how to resolve the resulting ambiguity when multiple root counts are compatible with the same sign pattern or when roots at zero or negative values affect the configuration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify our manuscript. We address the major comments point by point below, indicating revisions where the presentation can be strengthened.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reduction of the eigenvalue-configuration problem to real-root counting of the constructed symmetric polynomials is asserted without derivation steps or verification that the counts uniquely encode the desired configuration for generic parametric matrices.

    Authors: The abstract summarizes the main contribution, while the derivation of the reduction—via the characteristic polynomials of the two symmetric matrices, transformation to symmetric polynomials in the eigenvalues, and application of the Fundamental Theorem of Symmetric Polynomials—is provided in full in Sections 2–4 of the manuscript. The uniqueness for generic parameters follows from the fact that the eigenvalue ordering on the real line is completely determined by the signs and multiplicities captured in the root counts of these polynomials. We agree the abstract could better indicate these steps and will revise it to include a concise outline of the algorithm. We will also add a short verification paragraph in the introduction confirming uniqueness for generic cases. revision: partial

  2. Referee: [Abstract] Abstract: Descartes' rule of signs is invoked to obtain the root counts, yet the rule supplies only an upper bound (number of sign changes or less by an even integer) on the number of positive real roots. The manuscript does not address how to resolve the resulting ambiguity when multiple root counts are compatible with the same sign pattern or when roots at zero or negative values affect the configuration.

    Authors: Descartes' rule is used in the manuscript to bound the number of positive real roots of the constructed symmetric polynomials. In the specific polynomials arising from the eigenvalue configuration problem, the sign patterns and the separate treatment of zero and negative roots (via constant terms and even/odd degree considerations) ensure that the possible root counts consistent with a given sign pattern correspond uniquely to distinct configurations. We acknowledge that this resolution is not explicitly discussed. We will revise the manuscript by adding a dedicated paragraph after the statement of the algorithm that explains how ambiguities are avoided in this setting and, where the bound is not sharp, how the configuration conditions remain well-defined. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external standard theorems

full rationale

The paper constructs an algorithm that reduces the eigenvalue configuration problem to counting real roots of explicitly derived symmetric polynomials in the parameters. It then invokes the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs—both independent, externally established results with no self-citation or internal definition—to obtain the counts. No equation or step equates a derived quantity to its own input by construction, renames a fitted result as a prediction, or loads the central claim on a self-citation chain. The mapping is presented as a direct algorithmic translation whose validity rests on the cited theorems rather than on quantities defined inside the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on two classical theorems whose validity is taken from the mathematical literature; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Fundamental Theorem of Symmetric Polynomials
    Invoked to express symmetric polynomials in terms of elementary symmetric sums so that root counting becomes feasible.
  • standard math Descartes' rule of signs
    Invoked to count or bound the number of positive real roots of the resulting polynomials.

pith-pipeline@v0.9.0 · 5606 in / 1273 out tokens · 22290 ms · 2026-05-24T05:01:11.080688+00:00 · methodology

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Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Emiris, and Raimundas Vidunas

    Evangelos Bartzos, Ioannis Z. Emiris, and Raimundas Vidunas. New upper bounds for the number of embeddings of minimally rigid graphs, 2021

    work page 2021

  2. [2]

    Ben-Or, D

    M. Ben-Or, D. Kozen, and J. H. Reif. The complexity of elementary algebra and geometry.J. Comput. System Sci., 32(2):251–264, 1986

    work page 1986

  3. [3]

    The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices.Advances in Mathematics, 227(1):494–521, 2011

    Florent Benaych-Georges and Raj Rao Nadakuditi. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices.Advances in Mathematics, 227(1):494–521, 2011

    work page 2011

  4. [4]

    C. W. Brown. Improved projection for cylindrical algebraic decomposition.J. Symb. Comput., 32(5):447–465, 2001

    work page 2001

  5. [5]

    C. W. Brown. Simple cad construction and its applications.J. Symb. Comput., 31(5):521–547, 2001

    work page 2001

  6. [6]

    C. W. Brown. Fast simplifications for tarski formulas based on monomial inequalities.J. Symb. Comput., 47(7):859–882, 2012

    work page 2012

  7. [7]

    J. F. Canny. Improved algorithms for sign and existential quantifier elimination.Computer Journal, 36:409–418, 1993. In a special issue oncomputational quantifier elimination, edited by H. Hong

    work page 1993

  8. [8]

    C. Chen, M. Moreno Maza, B. Xia, and L. Yang. Computing cylindrical algebraic decomposition via triangular decomposition. InISSAC, pages 95–102, 2009

    work page 2009

  9. [9]

    G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination.Journal of Symbolic Computation, 12(3):299–328, sep 1991

    work page 1991

  10. [10]

    Existence of a unique solution to parametrized systems of generalized polynomial equations, 2024

    Abhishek Deshpande and Stefan M¨ uller. Existence of a unique solution to parametrized systems of generalized polynomial equations, 2024

    work page 2024

  11. [11]

    The hermitian killing form and root counting of complex polynomials with conjugate variables.Linear Algebra and its Applications, 708:93–111, March 2025

    Davide Furch` ı. The hermitian killing form and root counting of complex polynomials with conjugate variables.Linear Algebra and its Applications, 708:93–111, March 2025

    work page 2025

  12. [12]

    Degree 6 hyperbolic polynomials and orders of moduli, 2023

    Yousra Gati, Vladimir Petrov Kostov, and Mohamed Chaouki Tarchi. Degree 6 hyperbolic polynomials and orders of moduli, 2023

    work page 2023

  13. [13]

    Gonzalez-Vega

    L. Gonzalez-Vega. A combinatorial algorithm solving some quantifier elimination problems. In B. Cavi- ness and J. Johnson, editors,Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, 1996. Texts and Monographs in Symbolic Computation

    work page 1996

  14. [14]

    Gonzalez-Vega, H

    L. Gonzalez-Vega, H. Lombardi, T. Recio, and M.-F. Roy. Sturm-Habicht sequences. InProceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebriaic Computation, pages 136–146, July 1989. 18

    work page 1989

  15. [15]

    D. Yu. Grigor’ev. The complexity of deciding Tarski algebra.Journal of Symbolic Computation, 5(1,2):65–108, 1988

    work page 1988

  16. [16]

    Refined interlacing properties.SIAM journal on matrix analysis and applications, 13(1):239–247, 1992

    RO Hill, Jr and Beresford N Parlett. Refined interlacing properties.SIAM journal on matrix analysis and applications, 13(1):239–247, 1992

    work page 1992

  17. [17]

    H. Hong. An improvement of the projection operator in cylindrical algebraic decomposition. InInter- national Symposium of Symbolic and Algebraic Computation (ISSAC-90), pages 261–264. ACM, 1990

    work page 1990

  18. [18]

    Hong.Improvements in CAD–based Quantifier Elimination

    H. Hong.Improvements in CAD–based Quantifier Elimination. PhD thesis, The Ohio State University, 1990

    work page 1990

  19. [19]

    H. Hong. Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination. InInternational Conference on Symbolic and Algebraic Computation ISSAC-92, pages 177–188, 1992

    work page 1992

  20. [20]

    Hong and M

    H. Hong and M. S. El Din. Variant quantifier elimination.J. Symb. Comput., 47(7):883–901, 2012

    work page 2012

  21. [21]

    H. Hong, D. Profili, and J. R. Sendra. Conditions for eigenvalue configurations of two real symmetric matrices.ACM Commun. Comput. Algebra, 58(3):72–76, February 2025

    work page 2025

  22. [22]

    Conditions for eigenvalue configurations of two real symmetric matrices: a signature approach.arXiv preprint arXiv:2401.00866, 2023

    Hoon Hong, Daniel Profili, and J Rafael Sendra. Conditions for eigenvalue configurations of two real symmetric matrices: a signature approach.arXiv preprint arXiv:2401.00866, 2023

    work page arXiv 2023

  23. [23]

    Loos and V

    R. Loos and V. Weispfenning. Applying linear quantifier elimination.Computer Journal, 36(5):450–462,

    work page

  24. [24]

    In a special issue oncomputational quantifier elimination, edited by H. Hong

    work page

  25. [25]

    McCallum.An Improved Projection Operator for Cylindrical Algebraic Decomposition

    S. McCallum.An Improved Projection Operator for Cylindrical Algebraic Decomposition. PhD thesis, University of Wisconsin-Madison, 1984

    work page 1984

  26. [26]

    McCallum

    S. McCallum. Factors of iterated resultants and discriminants.J. Symb. Comput., 27(4):367–385, 1999

    work page 1999

  27. [27]

    McCallum and C

    S. McCallum and C. W. Brown. On delineability of varieties in cad-based quantifier elimination with two equational constraints. InISSAC, pages 71–78, 2009

    work page 2009

  28. [28]

    On one- parameter families of hermiticity-preserving superoperators which are not positive, 2024

    Grzegorz Pastuszak, Alicja Jaworska-Pastuszak, Takeo Kamizawa, and Andrzej Jamio lkowski. On one- parameter families of hermiticity-preserving superoperators which are not positive, 2024

    work page 2024

  29. [29]

    J. Renegar. On the computational complexity and geometry of the first-order theory of the reals.Journal of Symbolic Computation, 13(3):255–352, 1992

    work page 1992

  30. [30]

    Strzebonski

    A. Strzebonski. Cylindrical algebraic decomposition using validated numerics.J. Symb. Comput., 41(9):1021–1038, 2006

    work page 2006

  31. [31]

    Springer, 2008

    Bernd Sturmfels.Algorithms in invariant theory. Springer, 2008

    work page 2008

  32. [32]

    M. L. Telek. Geometry of the signed support of a multivariate polynomial and descartes’ rule of signs. SIAM Journal on Applied Algebra and Geometry, 8(4):968–1000, 2024

    work page 2024

  33. [33]

    Weispfenning

    V. Weispfenning. Quantifier elimination for real algebra – the cubic case. InInternational Symposium on Symbolic and Algebraic Computation 94, pages 258–263, 1994

    work page 1994

  34. [34]

    Weispfenning

    V. Weispfenning. A new approach to quantifier elimination for real algebra. In B. Caviness and J. Johnson, editors,Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag,

    work page

  35. [35]

    Acknowledgements.Hoon Hong was partially supported by US National Science Foundation NSF-CCF- 2212461

    Texts and Monographs in Symbolic Computation. Acknowledgements.Hoon Hong was partially supported by US National Science Foundation NSF-CCF- 2212461. 19

    work page