# zbMATH — the first resource for mathematics

Livsic-type determinantal representations and hyperbolicity. (English) Zbl 1391.32009
Summary: Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety $$X \subset \mathbb{P}^d$$ of an arbitrary codimension $$\ell$$ with respect to a real $$\ell - 1$$-dimensional linear subspace $$V \subset \mathbb{P}^d$$ and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces $$(\ell = 1)$$, the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in $$\mathbb{P}^d$$ hyperbolic with respect to some real $$d - 2$$-dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.

##### MSC:
 32C05 Real-analytic manifolds, real-analytic spaces 14M12 Determinantal varieties 30F10 Compact Riemann surfaces and uniformization
Macaulay2
Full Text:
##### References:
 [1] Ahlfors, L. L., Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv., 24, 100-134, (1950) · Zbl 0041.41102 [2] Ahlfors, L. V., Bounded analytic functions, Duke Math. J., 14, 1-11, (1947) · Zbl 0030.03001 [3] Alpay, D.; Vinnikov, V., Indefinite Hardy spaces on finite bordered Riemann surfaces, J. Funct. Anal., 172, 1, 221-248, (2000) · Zbl 0952.30035 [4] Alpay, D.; Vinnikov, V., Finite dimensional de branges spaces on Riemann surfaces, J. Funct. Anal., 189, 2, 283-324, (2002) · Zbl 1038.46020 [5] J.A. Ball, V. Vinnikov, Discrete-time 2D overdetermined linear systems: system-theoretic properties and transfer-function Hankel realization for meromorphic bundle maps on a compact Riemann surface, in preparation. [6] Ball, J. A.; Vinnikov, V., Zero-pole interpolation for meromorphic matrix functions on an algebraic curve and transfer functions of 2D systems, Acta Appl. Math., 45, 3, 239-316, (1996) · Zbl 0861.47010 [7] Ball, J. A.; Vinnikov, V., Zero-pole interpolation for matrix meromorphic functions on a compact Riemann surface and a matrix fay trisecant identity, Amer. J. Math., 121, 4, 841-888, (1999) · Zbl 0945.47008 [8] Bauschke, H. H.; Güler, O.; Lewis, A. S.; Sendov, H. S., Hyperbolic polynomials and convex analysis, Canad. J. Math., 53, 3, 470-488, (2001) · Zbl 0974.90015 [9] Beauville, A., Determinantal hypersurfaces, Michigan Math. J., 48, 39-64, (2000), Dedicated to William Fulton on the occasion of his 60th birthday · Zbl 1076.14534 [10] (Blekherman, G.; Parrilo, P. A.; Thomas, R. R., Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Ser. Optim., vol. 13, (2013), Society for Industrial and Applied Mathematics (SIAM)/Mathematical Optimization Society Philadelphia, PA) · Zbl 1260.90006 [11] Brändén, P., Obstructions to determinantal representability, Adv. Math., 226, 2, 1202-1212, (2011) · Zbl 1219.90121 [12] Busemann, H., Convexity on Grassmann manifolds, Enseign. Math. (2), 7, 139-152, (1962), 1961 · Zbl 0104.16902 [13] Chow, W. L.; van der Waerden, B. L., Uber zugeordenere formen und algebraische systeme von algebraischen mannifaltigkeiten, Math. Ann., 113, 692-704, (1937) · Zbl 0016.04004 [14] Dickson, L. E., Determination of all general homogeneous polynomials expressible as determinants with linear elements, Trans. Amer. Math. Soc., 22, 2, 167-179, (1921) · JFM 48.0099.02 [15] Dixon, A., Note on the reduction of a ternary quartic to a symmetrical determinant, Proc. Cambridge Philos. Soc., 11, 350-351, (1900-1902) · JFM 33.0140.04 [16] Dubrovin, B. A., Matrix finite zone operators, Contemp. Probl. Math., 23, 33-78, (1983) [17] Fay, J. D., Theta functions on Riemann surfaces, Lecture Notes in Math., vol. 352, (1973), Springer-Verlag Berlin · Zbl 0281.30013 [18] Fulton, W., Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, (1998), Springer-Verlag Berlin · Zbl 0885.14002 [19] Gårding, L., Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85, 1-62, (1951) · Zbl 0045.20202 [20] Gȧrding, L., An inequality for hyperbolic polynomials, J. Math. Mech., 8, 957-965, (1959) · Zbl 0090.01603 [21] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, resultants and multidimensional determinants, (2008), Modern Birkhäuser Classics. Birkhäuser Boston, Inc. Boston, MA, Reprint of the 1994 edition · Zbl 1138.14001 [22] Gohberg, I.; Lancaster, P.; Rodman, L., Indefinite linear algebra and applications, (2005), Birkhäuser Verlag Basel · Zbl 1084.15005 [23] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, available at [24] Güler, O., Hyperbolic polynomials and interior point methods for convex programming, Math. Oper. Res., 22, 2, 350-377, (1997) · Zbl 0883.90099 [25] Gustafsson, B.; Tkachev, V. G., The resultant on compact Riemann surfaces, Comm. Math. Phys., 286, 1, 313-358, (2009) · Zbl 1191.30015 [26] Gustafsson, B.; Tkachev, V. G., On the exponential transform of multi-sheeted algebraic domains, Comput. Methods Funct. Theory, 11, 2, 591-615, (2011) · Zbl 1246.30072 [27] Harris, J., Algebraic geometry, Grad. Texts in Math., vol. 133, (1995), Springer-Verlag New York, A first course, Corrected reprint of the 1992 original [28] Helton, J. W.; Vinnikov, V., Linear matrix inequality representation of sets, Comm. Pure Appl. Math., 60, 5, 654-674, (2007) · Zbl 1116.15016 [29] Kerner, D.; Vinnikov, V., Determinantal representations of singular hypersurfaces in $$\mathbb{P}^n$$, Adv. Math., 231, 3-4, 1619-1654, (2012) · Zbl 1251.14036 [30] Kravitsky, N., Discriminant varieties and discriminant ideals for operator vessels in Banach space, Integral Equations Operator Theory, 23, 4, 441-458, (1995) · Zbl 0854.47007 [31] Kummer, M., Determinantal representations and the Bézout matrix, (2014) [32] Lax, P. D., Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math., 11, 175-194, (1958) · Zbl 0086.01603 [33] Lewis, A. S.; Parrilo, P. A.; Ramana, M. V., The Lax conjecture is true, Proc. Amer. Math. Soc., 133, 9, 2495-2499, (2005), (electronic) · Zbl 1073.90029 [34] Livšic, M. S.; Kravitsky, N.; Markus, A. S.; Vinnikov, V., Theory of commuting nonselfadjoint operators, Math. Appl., vol. 332, (1995), Kluwer Academic Publishers Group Dordrecht · Zbl 0834.47004 [35] Markus, A.; Spielman, D. A.; Srivastava, N., Interlacing families ii: mixed characteristic polynomials and the kadison-Singer problem, (2014) [36] Nemirovski, A., Advances in convex optimization: conic programming, (International Congress of Mathematicians, vol. I, (2007), Eur. Math. Soc. Zürich), 413-444 · Zbl 1135.90379 [37] Nesterov, Y.; Nemirovskii, A., Interior-point polynomial algorithms in convex programming, SIAM Stud. Appl. Math., vol. 13, (1994), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0824.90112 [38] Ramana, M.; Goldman, A. J., Some geometric results in semidefinite programming, J. Global Optim., 7, 1, 33-50, (1995) · Zbl 0839.90093 [39] Renegar, J., Hyperbolic programs, and their derivative relaxations, Found. Comput. Math., 6, 1, 59-79, (2006) · Zbl 1130.90363 [40] Shapiro, A., Elimination theory on an algebraic curve and rational transformations of commuting nonselfadjoint operators, (1999), Weizmann Institute of Science, PhD thesis [41] Shapiro, A.; Vinnikov, V., Rational transformation of commuting nonselfadjoint operators, (2005) [42] Shapiro, A.; Vinnikov, V., Rational transformations of algebraic curves and elimination theory, (2005) [43] Vandenberghe, L.; Boyd, S., Semidefinite programming, SIAM Rev., 38, 1, 49-95, (1996) · Zbl 0845.65023 [44] Vinnikov, V., Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl., 125, 103-140, (1989) · Zbl 0704.14041 [45] Vinnikov, V., Selfadjoint determinantal representations of real plane curves, Math. Ann., 296, 3, 453-479, (1993) · Zbl 0789.14029 [46] Vinnikov, V., LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future, (Mathematical Methods in Systems, Optimization, and Control, Oper. Theory Adv. Appl., vol. 222, (2012), Birkhäuser/Springer Basel AG, Basel), 325-349 · Zbl 1276.14069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.