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Determinantal representation of trigonometric polynomial curves via Sylvester method. (English) Zbl 1283.15025
Let, for an \(n \times n\) complex matrix \(A\), \(\; F_A(t,x,y) := \det(t \, I_n+x \, \mathfrak{R}(A)+y \, \mathfrak{I}(A))\) denote the real ternary form associated with the matrix \(A\), and let \[ \phi(\theta) = \sum_{k=-n}^n c_k \exp (ik\theta) \] be a trigonometric polynomial. Using Sylvester matrices, these authors present an algorithm to construct \(2n \times 2n\) matrices \(C_1\), \(C_2\), \(C_3\) such that, for a given trigonometric polynomial \(\phi(\theta),\) \[ \det (C_1 + \mathfrak{R}(\phi(\theta)) C_2 + \mathfrak{I} (\phi(\theta)) C_3) = 0. \] If \(\phi\) has the special form \(\phi(\theta) = \exp(in\theta) + a \exp(-im\theta)\), \(0 \leq \theta \leq 2\pi\), \(0 < m < n\), \(0 < a < 1\), then the matrices \(C_1\), \(C_2\), \(C_3\) are Hermitian, \(C_1\) is positive definite, and the relation \[ F_{C_0}(t,x,y) \det(C_1) = \det(tC_1+xC_2+yC_3), \] holds, where \(C_0 = C_1^{-1/2} (C_2+iC_3) C_1^{-1/2}.\)

MSC:
15A15 Determinants, permanents, traces, other special matrix functions
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A12 Numerical range, numerical radius
42A05 Trigonometric polynomials, inequalities, extremal problems
15B57 Hermitian, skew-Hermitian, and related matrices
15B05 Toeplitz, Cauchy, and related matrices
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References:
[1] E. Bézout, General Theory of Algebraic Equations , Princeton University Press, Princeton and Oxford, 2006. (Tralslared from the French by E. Feron)
[2] M.T. Chien and H. Nakazato, Numerical range for orbits under a central force , Math. Phys. Anal. Geom. 13 (2010), 315-330. · Zbl 1245.15023 · doi:10.1007/s11040-010-9082-y
[3] M.T. Chien and H. Nakazato, Construction of determinantal representation of trigonometric polynomials , Linear Algebra Appl. 435 (2011), 1277-1284. · Zbl 1222.15008 · doi:10.1016/j.laa.2011.03.012
[4] M.T. Chien, H. Nakazato and P. Psarrakos, Point equation of the boundary of the numerical range of a matrix polynomial , Linear Algebra Appl. 347 (2002), 205-217. · Zbl 1006.15031 · doi:10.1016/S0024-3795(01)00549-3
[5] M. Fiedler, Geometry of the numerical range of matrices , Linear Algebra Appl. 37 (1981), 81-96. · Zbl 0452.15024 · doi:10.1016/0024-3795(81)90169-5
[6] J.W. Helton and V. Vinnikov, Linear matrix inequality representations of sets , Comm. Pure Appl. Math. 60 (2007), 654-674. · Zbl 1116.15016 · doi:10.1002/cpa.20155
[7] D. Henrion, Detecting rigid convexity of bivariate polynomials , Linear Algebra Appl. 432 (2010), 1218-1233. · Zbl 1183.65023 · doi:10.1016/j.laa.2009.10.033
[8] R. Kippenhahn, Über den wertevorrat einer Matrix , Math. Nachr. 6(1951), 193-228. · Zbl 0044.16201 · doi:10.1002/mana.19510060306
[9] P.D. Lax, Differential equations, difference equations and matrix theory , Comm. Pure Appl. Math. 6 (1958), 175-194. · Zbl 0086.01603 · doi:10.1002/cpa.3160110203
[10] A.S. Lewis, P.A. Parrilo and M.V. Ramana, The Lax conjecture is true , Proc. Amer. Math. Soc. 133 (2005), 2495-2499. · Zbl 1073.90029 · doi:10.1090/S0002-9939-05-07752-X
[11] M. Mignotte, Mathematics for Computer Algebra , Springer, New York, 1992. · Zbl 0741.11002
[12] D. Plaumann, B. Sturmfels, and C. Vinzant, Computing linear matrix representations of Helton-Vinnikov curves , arXiv: · Zbl 1328.14093 · doi:10.1007/978-3-0348-0411-0_19 · arxiv.org
[13] R.J. Walker, Algebraic curves , Dover Publ., New York, 1950. · Zbl 0039.37701
[14] H.K. Wimmer, On the history of the Bezoutian and the resultant matrix , Linear Algebra Appl. 128 (1990), 27-34. · Zbl 0711.15028 · doi:10.1016/0024-3795(90)90280-P
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