Skip to main content
SpringerLink
Log in

On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

It is shown that the polynomial

p(t) = Tr[(A+tB)m]

has positive coefficients when m = 6 and A and B are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa, and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to p(t) having positive coefficients for any m and any two n-by-n positive definite matrices. We show that, generally, the question in the real case reduces to that of singular A and B, and this is a key part of our proof.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • D. Bessis P. Moussa M. Villani (1975) ArticleTitleMonotonic converging variational approximations to the functional integrals in quantum statistical mechanics J. Math. Phys. 16 2318–2325

    Google Scholar 

  • D. Cox J. Little D. O’Shea (1998) Using Algebraic Geometry Springer New York

    Google Scholar 

  • C. Hillar C. R. Johnson (2004) ArticleTitleSymmetric word equations in two positive definite letters Proc. Ame. Math. Soc. 132 945–953

    Google Scholar 

  • C. Hillar C. R. Johnson I. M. Spitkovsky (2002) ArticleTitlePositive eigenvalues and two-letter generalized words, Electron. J. Linear Algebra 9 21–26

    Google Scholar 

  • R. Horn C. R. Johnson (1985) Matrix Analysis Cambridge University Press New York

    Google Scholar 

  • R. Horn C. R. Johnson (1991) Topics in Matrix Analysis Cambridge University Press New York

    Google Scholar 

  • C. R. Johnson C. Hillar (2002) ArticleTitleEigenvalues of Words in Two Positive Definite Letters SIAM J. Matrix Anal. Appl. 23 916–928

    Google Scholar 

  • E. H. Lieb R. Seiringer (2004) ArticleTitleEquivalent forms of the Bessis–Moussa–Villani conjecture J. Stat. Phys. 115 185–190

    Google Scholar 

  • B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC, (American Mathematical Society, Providence, RI, 2002).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Christopher J. Hillar

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Charles R. Johnson

Authors
  1. Christopher J. Hillar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles R. Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles R. Johnson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hillar, C.J., Johnson, C.R. On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices. J Stat Phys 118, 781–789 (2005). https://doi.org/10.1007/s10955-004-8829-x

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-004-8829-x

Keywords

Search

Navigation