×

Poncelet-Darboux, Kippenhahn, and Szegő: interactions between projective geometry, matrices and orthogonal polynomials. (English) Zbl 1486.14047

The authors study algebraic curves that are envelopes of families of polygons supported on the unit circle \(\mathbb{T}\) and look for a characterization of such curves of minimal class. All realizations of these curves are shown to be essentially equivalent. They can be described in terms of Szegö polynomials, i.e. orthogonal polynomials on the unit circle. The results of the paper are related to theorems of Poncelet, Darboux, and Kippenhahn, to Blaschke products and disk functions etc. The authors give a rigorous definition of the Poncelet property, of curves tangent to a family of polygons, and of polygons associated with Poncelet curves. This is illustrated by detailed figures. Some misconceptions that appear in the literature are clarified and counterexamples to some existing assertions are presented. Example: curves inscribed in some families of polygons supported on \(\mathbb{T}\) are not necessarily convex, can have cusps and can even intersect the unit circle. The authors show the usefulness of Szegö polynomials and of tangent coordinates; the latter idea has been previously exploited by B. Mirman.

MSC:

14H50 Plane and space curves
14P05 Real algebraic sets
51N35 Questions of classical algebraic geometry
30J10 Blaschke products
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andrews, G. E.; Dragović, V.; Radnovic, M., Combinatorics of periodic ellipsoidal billiards, Ramanujan J. (2021)
[2] Beltrametti, M. C.; Carletti, E.; Gallarati, D.; Monti Bragadin, G., A classical view of algebraic geometry, (Lectures on Curves, Surfaces and Projective Varieties. Lectures on Curves, Surfaces and Projective Varieties, EMS Textbooks in Mathematics (2009), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich), Translated from the 2003 Italian original by Francis Sullivan · Zbl 1180.14001
[3] Berger, M., Geometry I, Universitext (2009), Springer-Verlag: Springer-Verlag Berlin, Translated from the 1977 French original by M. Cole and S. Levy, Fourth printing of the 1987 English translation · Zbl 1153.51001
[4] Brown, E. S.; Spitkovsky, I. M., On matrices with elliptical numerical ranges, Linear Multilinear Algebra, 52, 3-4, 177-193 (May 2004) · Zbl 1059.15030
[5] Burskii, V. P.; Zhedanov, A. S., On Dirichlet, Poncelet and Abel problems, Commun. Pure Appl. Anal., 12, 4, 1587-1633 (July 2013) · Zbl 1264.14043
[6] Chien, M.-T.; Hung, K.-C., Elliptic numerical ranges of bordered matrices, Taiwan. J. Math., 16, 3, 1007-1016 (May 2012) · Zbl 1259.15035
[7] Daepp, U.; Gorkin, P.; Mortini, R., Ellipses and finite Blaschke products, Am. Math. Mon., 109, 9, 785-795 (Nov. 2002) · Zbl 1022.30039
[8] Daepp, U.; Gorkin, P.; Shaffer, A.; Sokolowsky, B.; Voss, K., Decomposing finite Blaschke products, J. Math. Anal. Appl., 426, 2, 1201-1216 (2015) · Zbl 1325.30058
[9] Daepp, U.; Gorkin, P.; Shaffer, A.; Voss, K., Möbius transformations and Blaschke products: the geometric connection, Linear Algebra Appl., 516, 186-211 (Mar. 2017) · Zbl 1380.30040
[10] Daepp, U.; Gorkin, P.; Shaffer, A.; Voss, K., Finding Ellipses. What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know About Each Other, Carus Mathematical Monographs, vol. 34 (2018), MAA Press: MAA Press Providence, RI · Zbl 1419.51001
[11] Daepp, U.; Gorkin, P.; Voss, K., Poncelet’s theorem, Sendov’s conjecture, and Blaschke products, J. Math. Anal. Appl., 365, 1, 93-102 (May 2010) · Zbl 1183.30062
[12] Darboux, G., Principes de géométrie analytique (1917), Gauthier-Villars: Gauthier-Villars Paris · JFM 46.0877.14
[13] Del Centina, A., Poncelet’s porism: a long story of renewed discoveries, I, Arch. Hist. Exact Sci., 70, 1, 1-122 (2016) · Zbl 1333.01015
[14] Del Centina, A., Poncelet’s porism: a long story of renewed discoveries, II, Arch. Hist. Exact Sci., 70, 2, 123-173 (2016) · Zbl 1345.01007
[15] Dragović, V., Poncelet-Darboux curves, their complete decomposition and Marden theorem, Int. Math. Res. Not., 2011, 15, 3502-3523 (Oct. 2011) · Zbl 1230.14038
[16] Dragović, V.; Radnovic, M., Bicentennial of the Great Poncelet Theorem (1813-2013): current advances, Bull. Am. Math. Soc., 51, 3, 373-445 (July 2014) · Zbl 1417.37034
[17] Dragović, V.; Radnovic, M., Caustics of Poncelet polygons and classical extremal polynomials, Regul. Chaotic Dyn., 24, 1, 1-35 (Feb. 2019) · Zbl 1432.37055
[18] Flatto, L., Poncelet’s Theorem (2009), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1157.51001
[19] Fujimura, M., Inscribed ellipses and Blaschke products, Comput. Methods Funct. Theory, 13, 4, 557-573 (Sept. 2013) · Zbl 1291.30040
[20] Fujimura, M., Blaschke products and circumscribed conics, Comput. Methods Funct. Theory, 17, 4, 635-652 (May 2017) · Zbl 1394.30045
[21] Fujimura, M., Interior and exterior curves of finite Blaschke products, J. Math. Anal. Appl., 467, 1, 711-722 (Nov. 2018) · Zbl 1405.30059
[22] Gao, S., Absolute irreducibility of polynomials via Newton polytopes, J. Algebra, 237, 2, 501-520 (2001) · Zbl 0997.12001
[23] Gau, H.-L., Elliptic numerical ranges of \(4 \times 4\) matrices, Taiwan. J. Math., 1, 117-128 (Jan. 2006) · Zbl 1099.15022
[24] Gau, H.-L.; Wu, P. Y., Dilation to inflations of \(S(\phi)\), Linear Multilinear Algebra, 45, 2-3, 109-123 (Dec. 1998) · Zbl 0943.47006
[25] Gau, H.-L.; Wu, P. Y., Numerical range of \(S(\phi)\), Linear Multilinear Algebra, 45, 1, 49-73 (Nov. 1998) · Zbl 0918.15008
[26] Gau, H.-L.; Wu, P. Y., Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl., 364, 213-222 (2003) · Zbl 1026.15018
[27] Gau, H.-L.; Wu, P. Y., Numerical range and Poncelet property, Taiwan. J. Math., 7, 2, 173-193 (2003) · Zbl 1051.15019
[28] Geronimus, J. L.; Sneddon, Ian N., Polynomials Orthogonal on a Circle and Interval, International Series of Monographs on Pure and Applied Mathematics, vol. 18 (1960), Pergamon Press: Pergamon Press New York-Oxford-London-Paris, Translated from the Russian by D.E. Brown
[29] Goldschmidt, D. M., Algebraic Functions and Projective Curves, Graduate Texts in Mathematics, vol. 215 (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1034.14011
[30] Gorkin, P., Four theorems with their foci on ellipses, Am. Math. Mon., 126, 2, 99-111 (Feb. 2019) · Zbl 1410.51029
[31] Gorkin, P.; Wagner, N., Ellipses and compositions of finite Blaschke products, J. Math. Anal. Appl., 445, 2, 1354-1366 (Jan. 2017) · Zbl 1354.30053
[32] Griffiths, P.; Harris, J., On Cayley’s explicit solution to Poncelet’s porism, Enseign. Math., 24, 31-40 (1978) · Zbl 0384.14009
[33] Haagerup, U.; de la Harpe, P., The numerical radius of a nilpotent operator on a Hilbert space, Proc. Am. Math. Soc., 115, 2, 371-379 (1992) · Zbl 0781.47014
[34] Halmos, P. R., A Hilbert Space Problem Book (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0202.12801
[35] Horn, R. A.; Johnson, C. R., Matrix Analysis (2013), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1267.15001
[36] Hunziker, M.; Martinez-Finkelshtein, A.; Poe, T.; Simanek, B., On foci of ellipses inscribed in cyclic polygons, in: F. Gesztesy, A. Martinez-Finkelshtein (Eds.), From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory, in: Operator Theory: Advances and Applications, vol. 285, pp. 213-238 · Zbl 1498.30032
[37] Kendig, K., A Guide to Plane Algebraic Curves, The Dolciani Mathematical Expositions, vol. 46 (2011), Mathematical Association of America: Mathematical Association of America Washington, DC · Zbl 1229.14001
[38] Kippenhahn, R., Über den Wertevorrat einer Matrix, Math. Nachr., 6, 193-228 (1951) · Zbl 0044.16201
[39] Kippenhahn, R., On the numerical range of a matrix, Linear Multilinear Algebra, 56, 1-2, 185-225 (2008), Translated from the German by Paul F. Zachlin and Michiel E. Hochstenbach · Zbl 1137.47003
[40] Kudryavtseva, E. A., Liouville integrable generalized billiard flows and theorems of Poncelet type, Fundam. Prikl. Mat., 20, 3, 113-152 (2015)
[41] Langer, J. C.; Singer, D. A., Foci and foliations of real algebraic curves, Milan J. Math., 75, 1, 225-271 (Nov. 2007) · Zbl 1150.14005
[42] Last, Y.; Simon, B., Fine structure of the zeros of orthogonal polynomials. IV. A priori bounds and clock behavior, Commun. Pure Appl. Math., 61, 4, 486-538 (2008) · Zbl 1214.42044
[43] Li, C.-K., A simple proof of the elliptical range theorem, Proc. Am. Math. Soc., 124, 7, 1985-1986 (1996) · Zbl 0857.15020
[44] Livshitz, M. S., On a certain class of linear operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S., 19, 61, 239-262 (1946) · Zbl 0061.25903
[45] Marden, M., Geometry of Polynomials, Math. Surveys, vol. 3 (1966), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I. · Zbl 0173.02703
[46] Martínez-Finkelshtein, A.; Simanek, B.; Simon, B., Poncelet’s theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators, Adv. Math., 349, 992-1035 (2019) · Zbl 1497.47012
[47] Mirman, B., UB-matrices and conditions for Poncelet polygon to be closed, Linear Algebra Appl., 360, 123-150 (2003) · Zbl 1028.15025
[48] Mirman, B., Sufficient conditions for Poncelet polygons not to close, Am. Math. Mon., 112, 4, 351-356 (Apr. 2005) · Zbl 1215.52001
[49] Mirman, B., Explicit solutions to Poncelet’s porism, Linear Algebra Appl., 436, 9, 3531-3552 (May 2012) · Zbl 1253.14052
[50] Mirman, B.; Shukla, P., A characterization of complex plane Poncelet curves, Linear Algebra Appl., 408, 86-119 (Oct. 2005) · Zbl 1076.14076
[51] L. Moret-Bailly, Private communication, 2021.
[52] Ovsienko, V.; Schwartz, R.; Tabachnikov, S., The pentagram map: a discrete integrable system, Commun. Math. Phys., 299, 2, 409-446 (June 2010) · Zbl 1209.37063
[53] Pecker, D., Poncelet’s theorem and billiard knots, Geom. Dedic., 161, 323-333 (2012) · Zbl 1282.57011
[54] Poncelet, J.-V., Traité sur les propriétés projectives des figures, (ProQuest LLC. ProQuest LLC, Ann Arbor, MI (1822))
[55] Richter-Gebert, J., Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry (2011), Springer: Springer Heidelberg · Zbl 1214.51001
[56] Schwartz, R. E., The pentagram map, Exp. Math., 1, 71-81 (Jan. 1992) · Zbl 0765.52004
[57] Schwartz, R. E., The pentagram integrals for Poncelet families, J. Geom. Phys., 87, 432-449 (Jan. 2015) · Zbl 1309.37068
[58] Shafarevich, I. R., Basic Algebraic Geometry. 1: Varieties in Projective Space (2013), Springer: Springer Heidelberg · Zbl 1273.14004
[59] Siebeck, J., Ueber eine neue analytische Behandlungweise der Brennpunkte, J. Reine Angew. Math., 64, 175-182 (1864) · ERAM 064.1672cj
[60] Simon, B., Orthogonal Polynomials on the Unit Circle I and II, AMS Colloquium Publications, vol. 54 (2005), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1082.42020
[61] Stoiciu, M., The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle, J. Approx. Theory, 139, 1-2, 29-64 (2006) · Zbl 1088.42017
[62] Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Kérchy, L., Harmonic Analysis of Operators on Hilbert Space, Universitext (2010), Springer: Springer New York · Zbl 1234.47001
[63] Szegő, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23 (1975), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · JFM 61.0386.03
[64] Tabachnikov, S., Kasner meets Poncelet, Math. Intell., 41, 4, 56-59 (2019) · Zbl 1458.51003
[65] Wu, P. Y., Polygons and numerical ranges, Am. Math. Mon., 107, 6, 528-540 (2000) · Zbl 0985.51016
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.