The Michigan Mathematical Journal

Determinantal representations and the Hermite matrix

Tim Netzer, Daniel Plaumann, and Andreas Thom

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Michigan Math. J., Volume 62, Issue 2 (2013), 407-420.

First available in Project Euclid: 10 June 2013

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Primary: 11C20: Matrices, determinants [See also 15B36] 11E25: Sums of squares and representations by other particular quadratic forms 14P10: Semialgebraic sets and related spaces
Secondary: 90C22: Semidefinite programming 90C25: Convex programming 52B99: None of the above, but in this section


Netzer, Tim; Plaumann, Daniel; Thom, Andreas. Determinantal representations and the Hermite matrix. Michigan Math. J. 62 (2013), no. 2, 407--420. doi:10.1307/mmj/1370870379.

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  • S. Basu, R. Pollack, and M.-F. Roy, Algorithms in real algebraic geometry, Algorithms Comput. Math., 10, Springer-Verlag, Berlin, 2003.
  • J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), 36, Springer-Verlag, Berlin, 1998.
  • C. W. Borchardt, Neue Eigenschaft der Gleichung, mit deren Hülfe man die seculären Störungen der Planeten bestimmt, J. Reine Angew. Math. 12 (1846), 38–45.
  • P. Brändén, Obstructions to determinantal representability, Adv. Math. 226 (2011), 1202–1212.
  • A. C. Dixon, Note on the reduction of a ternary quantic to a symmetrical determinant, Math. Proc. Cambridge Philos. Soc. (5) 11 (1902), 350–351.
  • D. Gondard and P. Ribenboim, Le 17e problème de Hilbert pour les matrices, Bull. Sci. Math. (2) 98 (1974), 49–56.
  • B. Grenet, E. Kaltofen, P. Koiran, and N. Portier, Symmetric determinantal representation of formulas and weakly skew circuits, Randomization, relaxation, and complexity in polynomial equation solving, Contemp. Math., 556, pp. 61–96, Amer. Math. Soc., Providence, RI, 2011.
  • J. W. Helton, S. McCullough, and V. Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006), 105–191.
  • J. W. Helton and V. Vinnikov, Linear matrix inequality representation of sets, Comm. Pure Appl. Math. 60 (2007), 654–674.
  • D. Henrion, Detecting rigid convexity of bivariate polynomials, Linear Algebra Appl. 432 (2010), 1218–1233.
  • V. A. Jakubovic, Factorization of symmetric matrix polynomials, Dokl. Akad. Nauk SSSR 194 (1970), 532–535.
  • D. G. Mead, Newton's identities, Amer. Math. Monthly 99 (1992), 749–751.
  • T. Netzer and A. Thom, Polynomials with and without determinantal representations, Linear Algebra Appl. 437 (2012), 1579–1595.
  • W. Nuij, A note on hyperbolic polynomials, Math. Scand. 23 (1968), 69–72.
  • C. Procesi, Positive symmetric functions, Adv. Math. 29 (1978), 219–225.
  • R. Quarez, Symmetric determinantal representation of polynomials, Linear Algebra Appl. 436 (2012), 3642–3660.
  • V. Vinnikov, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl. 125 (1989), 103–140.