Distinguished varieties

Agler, Jim; McCarthy, John E.

doi:10.1007/bf02393219

Cited by 121 publications

(197 citation statements)

References 20 publications

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“…Having a solvable N -Pick problem on D n denote by V the set of points in D n on which all the interpolating functions have the same value. This set is usually a proper subset of D n and the argument of Agler and McCarthy [5] shows that V is a variety, what justifies its name. Agler and McCarthy [4] described uniqueness varieties for the three-Pick problem in D 2 .…”

Section: Introductionmentioning

confidence: 99%

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Three-point Nevanlinna–Pick problem in the polydisc

Kosiński¹

2015

Proc. London Math. Soc.

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Abstract. It is very elementary to observe that functions interpolating an extremal two-point Pick problem on the polydisc are just left inverses to complex geodesics.In the present article we show that the same property holds for a three-point Pick problem on polydiscs, i.e. it may be expressed it in terms of three-complex geodesics. Using this idea we are able to solve that problem obtaining formulas and a uniqueness theorem for solutions of extremal problems. In particular, we determine a class of rational inner functions interpolating that problem.Possible extensions and further investigations are also discussed.

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Section: Introductionmentioning

confidence: 99%

“…Agler and McCarthy [4] described uniqueness varieties for the three-Pick problem in D 2 . In [5] they showed that any uniqueness variety contains a distinguished variety W (that is a variety of the form W = {(z, w) ∈ D 2 : p(z, w) = 0} for some polynomial p and such that W ∩ ∂(D 2 ) = W ∩ T 2 ) that contains each of nodes.…”

Section: Introductionmentioning

confidence: 99%

Three-point Nevanlinna–Pick problem in the polydisc

Kosiński¹

2015

Proc. London Math. Soc.

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“…We view this as an analogue of the celebrated work of Agler and M c Carthy on norm preserving extensions of functions on varieties to whole domains [1,2]. Additionally, since p totally determinesp, the dimension of the Zariski closure of the image of a free polynomial map can apparently go up, in contrast with the commutative case [12].…”

Section: Geometrymentioning

confidence: 87%

Free functions with symmetry

2017

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In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf's theorem is a special case of a general theory of the ring of invariant free polynomials: every ring of invariant free polynomials is isomorphic to a free polynomial ring. Furthermore, we show that this isomorphism extends to the free functional calculus as a norm-preserving isomorphism of function spaces on a domain known as the row ball. We give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures. Specifically, we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case.

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“…This means that Φ holizes Ω as a distinguished variety. (The fact that φ 1 and φ 2 satisfy an algebraic equation was proved by Fedorov by extending them to the Schottky double, but also follows from general arguments [3].) However, W. Rudin showed that if the connectivity of Ω is greater than one, then Φ cannot be one-to-one on ∂Ω [22].…”

Section: Pairs Of Blaschke Productsmentioning

confidence: 94%

“…These are called distinguished varieties (since they exit the disk through the distinguished boundary) and have been studied in [3] and [27]. One difficulty in the theory is proving that a particular pair of functions in a cofinite algebra generate the whole algebra.…”

Section: Theorem 12 (I) If H Is a Holomap From A Riemann Surface S Imentioning

confidence: 99%

Hyperbolic algebraic and analytic curves

Agler¹,

McCarthy²

2007

Indiana Univ. Math. J.

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A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the holomorphic functions on the desingularization. We show that classical analytic techniques, such as interpolation, can be used to answer geometric questions about the existence of biholomorphic maps. Conversely, we show that the algebraic-geometric viewpoint leads to interesting questions in classical analysis. IntroductionBy a hyperbolic algebraic curve we mean a set V that is the intersection of a one dimensional algebraic set C with a bounded open set in C n . We call them hyperbolic to emphasize that we wish to study the geometry and function theory on V , not the global theory on C, nor just the local theory on small neighborhoods of points of V . Note that by a holomorphic function on V we mean a function that in a neighborhood of every point of V is the restriction of a holomorphic function on a neighborhood in C n .

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Distinguished varieties

Cited by 121 publications

References 20 publications

Three-point Nevanlinna–Pick problem in the polydisc

Three-point Nevanlinna–Pick problem in the polydisc

Free functions with symmetry

Hyperbolic algebraic and analytic curves

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