Convergence sets of a formal power series

Levenberg, Norman; Molzon, Robert

doi:10.1007/bf01418339

Cited by 19 publications

(18 citation statements)

References 5 publications

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“…Our main result states that a countable union of closed complete pluripolar sets in P n belongs to Conv(P n ). This generalizes the results of Lelong [9], Levenberg and Molzon [10], and Sathaye [16]. Our line of approach was inspired by [16], and influenced by the methods developed in [10,13,15].…”

supporting

confidence: 68%

“…Proof The proposition is proved by following the approach of [10,Theorem 5.6]. Let Then Since K is a compact pluripolar set, K � = P n (see [6,Corollary 4.4]).…”

Section: Proposition 44mentioning

confidence: 99%

“…Remark Proposition 4.4 is motivated by Theorem 5.6 in [10], which states that if U is an affine open set in P n and if K is a compact complete pluripolar set in U then K is the intersection of U and a convergence set in P n . We observe that the same reasoning which proves that theorem can be applied to prove the more general statement Proposition 4.4.…”

Section: Proposition 44mentioning

confidence: 99%

“…We observe that the same reasoning which proves that theorem can be applied to prove the more general statement Proposition 4.4. Proposition 4.4 is more general than Theorem 5.6 in [10] because (a) K may be noncompact, (b) K is not necessarily a complete pluripolar set, and (c) K does not have to lie in an affine open set.…”

Section: Proposition 44mentioning

confidence: 99%

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On convergence sets of formal power series

Neelon

2015

Complex Analysis and its Synergies

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BackgroundA formal power series f (x 0 , x 1 , . . . , x n ) with coefficients in C is said to be convergent if it converges absolutely in a neighborhood of the origin in C n+1 . A classical result of Hartogs (see [5]) states that a series f converges if and only if f z (t) := f (z 0 t, z 1 t, . . . , z n t) converges, as a series in t, for all z ∈ C n+1 . This can be interpreted as a formal analog of Hartogs' theorem on separate analyticity. Because a divergent power series still may converge in certain directions, it is natural and desirable to consider the set of all z ∈ C n+1 for which f z converges. Since f z (t) converges if and only if f w (t) converges for all w ∈ C n+1 on the affine line through z, ignoring the trivial case z = 0, the set of directions along which f converges can be identified with a subset of P n . The convergence set Conv(f ) of a divergent power series f is defined to be the set of all directions ξ ∈ P n such that f z (t) is convergent for some z ∈ π −1 (ξ ), where π : C n+1 \{0} → P n is the natural projection. For the case n = 1, Lelong [9] proved that the convergence set of a divergent series f (x 1 , x 2 ) is an F σ polar set (i.e. a countable union of closed sets of vanishing logarithmic capacity) in P 1 , and moreover, every F σ polar subset of P 1 is contained in the convergence set of a divergent series f (x 1 , x 2 ). The optimal result was later obtained by Sathaye (see [16]) who showed that the class of convergence sets of divergent power series f (x 1 , x 2 ) is precisely the class of F σ polar sets in P 1 . To study the collection Conv(P n ) of convergence sets of divergent series in higher dimensions we consider the class PSH ω (P n ) of ω-plurisubharmonic functions on P n with respect to the form ω := dd c log |Z| on P n . We show that Conv(P n ) contains projective hulls of AbstractThe convergence set of a divergent formal power series f (x 0 , . . . , x n ) is the set of all "directions" ξ ∈ P n along which f is absolutely convergent. We prove that every countable union of closed complete pluripolar sets in P n is the convergence set of some divergent series f. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional analogs of the results of Sathaye

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supporting

confidence: 68%

“…Proof The proposition is proved by following the approach of [10,Theorem 5.6]. Let Then Since K is a compact pluripolar set, K � = P n (see [6,Corollary 4.4]).…”

Section: Proposition 44mentioning

confidence: 99%

Section: Proposition 44mentioning

confidence: 99%

Section: Proposition 44mentioning

confidence: 99%

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On convergence sets of formal power series

Neelon

2015

Complex Analysis and its Synergies

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“…As mentioned above, a lot of work has been done on the convergence of a power series with the assumption that the series is convergent after restriction to sufficiently many subspaces (see [AM,LM,Le,Si,Si2]). …”

Section: Introductionmentioning

confidence: 99%

Osgood–Hartogs-type properties of power series and smooth functions

Fridman¹,

Ma²

2011

Pacific J. Math.

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Abstract. We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given C ∞ function f is proved when the restriction of f on analytic curves belonging to some family is analytic. Our results generalize two known statements: a theorem of P. Lelong and the Bochnak-Siciak Theorem. The questions we study fall into the category of "Osgood-Hartogstype" problems.

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Note on convergence and orders of vanishing along curves

Izumi

1990

Manuscripta Math

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Convergence sets of a formal power series

Cited by 19 publications

References 5 publications

On convergence sets of formal power series

On convergence sets of formal power series

Osgood–Hartogs-type properties of power series and smooth functions

Note on convergence and orders of vanishing along curves

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