Document Open Access Logo

Real Stability Testing

Authors Prasad Raghavendra, Nick Ryder, Nikhil Srivastava

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2017.5.pdf
  • Filesize: 0.5 MB
  • 15 pages

Document Identifiers

Author Details

Prasad Raghavendra
Nick Ryder
Nikhil Srivastava

Cite AsGet BibTex

Prasad Raghavendra, Nick Ryder, and Nikhil Srivastava. Real Stability Testing. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ITCS.2017.5

Abstract

We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials preserves real-rootedness. The proof exploits properties of hyperbolic polynomials to reduce real stability testing to testing nonnegativity of a finite number of polynomials on an interval.
Keywords
  • real stable polynomials
  • hyperbolic polynomials
  • real rootedness
  • moment matrix
  • sturm sequence

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Nima Anari and Shayan Oveis Gharan. The kadison-singer problem for strongly rayleigh measures and applications to asymmetric tsp. arXiv preprint arXiv:1412.1143, 2014. Google Scholar
  2. Saugata Basu, Richard Pollack, and Marie-Francoise Roy. Algorithms in real algebraic geometry, volume 20033. Springer, 2005. Google Scholar
  3. Michael Ben-Or, Dexter Kozen, and John Reif. The complexity of elementary algebra and geometry. Journal of Computer and System Sciences, 32(2):251-264, 1986. Google Scholar
  4. Julius Borcea and Petter Brändén. The lee-yang and pólya-schur programs. i. linear operators preserving stability. Inventiones mathematicae, 177(3):541-569, 2009. Google Scholar
  5. Julius Borcea, Petter Brändén, and Thomas Liggett. Negative dependence and the geometry of polynomials. Journal of the American Mathematical Society, 22(2):521-567, 2009. Google Scholar
  6. Lars Garding. An inequality for hyperbolic polynomials. Journal of Mathematics and Mechanics, 8(6):957-965, 1959. Google Scholar
  7. Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. A randomized rounding approach to the traveling salesman problem. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 550-559. IEEE, 2011. Google Scholar
  8. J William Helton and Victor Vinnikov. Linear matrix inequality representation of sets. Communications on pure and applied mathematics, 60(5):654-674, 2007. Google Scholar
  9. Didier Henrion. Detecting rigid convexity of bivariate polynomials. Linear Algebra and its Applications, 432(5):1218-1233, 2010. Google Scholar
  10. Roger A Horn and Charles R Johnson. Matrix analysis. Cambridge university press, 2012. Google Scholar
  11. Walter Keller-Gehrig. Fast algorithms for the characteristics polynomial. Theoretical computer science, 36:309-317, 1985. Google Scholar
  12. Mario Kummer, Daniel Plaumann, and Cynthia Vinzant. Hyperbolic polynomials, interlacers, and sums of squares. Mathematical Programming, 153(1):223-245, 2015. Google Scholar
  13. Adam Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families i: Bipartite ramanujan graphs of all degrees. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 529-537. IEEE, 2013. Google Scholar
  14. Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families ii: Mixed characteristic polynomials and the kadison-singer problem. Annals of Mathematics, 182(1):327-350, 2015. Google Scholar
  15. Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families iv: Bipartite ramanujan graphs of all sizes. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 1358-1377. IEEE, 2015. Google Scholar
  16. Wim Nuij. A note on hyperbolic polynomials. Mathematica Scandinavica, 23(1):69-72, 1969. Google Scholar
  17. Robin Pemantle. Hyperbolicity and stable polynomials in combinatorics and probability. arXiv preprint arXiv:1210.3231, 2012. Google Scholar
  18. Helfried Peyrl and Pablo A Parrilo. Computing sum of squares decompositions with rational coefficients. Theoretical Computer Science, 409(2):269-281, 2008. Google Scholar
  19. Charles Sturm. Mémoire sur la résolution des équations numériques. 1835. Google Scholar
  20. David Wagner. Multivariate stable polynomials: theory and applications. Bulletin of the American Mathematical Society, 48(1):53-84, 2011. Google Scholar
  21. David Y.Y. Yun. On square-free decomposition algorithms. In Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation, SYMSAC'76, pages 26-35, New York, NY, USA, 1976. ACM. URL: http://dx.doi.org/10.1145/800205.806320.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact