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Cyclic polynomials in two variables


Authors: Catherine Bénéteau, Greg Knese, Łukasz Kosiński, Constanze Liaw, Daniel Seco and Alan Sola
Journal: Trans. Amer. Math. Soc. 368 (2016), 8737-8754
MSC (2010): Primary 32A37, 47A13; Secondary 14M99
DOI: https://doi.org/10.1090/tran6689
Published electronically: February 12, 2016
MathSciNet review: 3551587
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Abstract: We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for polynomials, and harmonic analysis on curves.


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Additional Information

Catherine Bénéteau
Affiliation: Department of Mathematics, University of South Florida, 4202 E. Fowler Avenue, Tampa, Florida 33620-5700
Email: cbenetea@usf.edu

Greg Knese
Affiliation: Department of Mathematics, Washington University in St. Louis, One Brookings Drive, Campus Box 1146, St. Louis, Missouri 63130-4899
Email: geknese@math.wustl.edu

Łukasz Kosiński
Affiliation: Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland – and – Département de mathématiques et de statistique, Pavillon AlexandreVachon, 1045 av. de la Médecine, Université Laval, Québec (Québec) G1V 0A6, Canada
Email: lukasz.kosinski@gazeta.pl

Constanze Liaw
Affiliation: Department of Mathematics and CASPER, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
Email: Constanze_Liaw@baylor.edu

Daniel Seco
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
Address at time of publication: Departament de Matemàtica Aplicada i Analisi, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: dseco@mat.uab.cat

Alan Sola
Affiliation: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Address at time of publication: Department of Mathematics, University of South Florida, 4202 E. Fowler Avenue, Tampa, Florida 33620-5700
Email: a.sola@statslab.cam.ac.uk

DOI: https://doi.org/10.1090/tran6689
Keywords: Cyclicity, Dirichlet-type spaces, bidisk, determinantal representations
Received by editor(s): November 4, 2014
Published electronically: February 12, 2016
Additional Notes: The second author was supported by NSF grant DMS-1363239
The third author was supported by NCN grant 2011/03/B/ST1/04758
The fourth author was partially supported by NSF grant DMS-1261687
The fifth author was supported by ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013 and MEC/MICINN Project MTM2011-24606
The sixth author acknowledges support from the EPSRC under grant EP/103372X/1
Article copyright: © Copyright 2016 American Mathematical Society