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On Convergence Sets of Formal Power Series Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 DOI 10.1186/s40627-015-0004-4 RESEARCH Open Access On convergence sets of formal power series Daowei Ma1* and Tejinder S. Neelon2 *Correspondence: [email protected] Abstract 1 Department The convergence set of a divergent formal power series f (x0, ..., xn) is the set of all of Mathematics, Wichita “directions” ξ Pn along which f is absolutely convergent. We prove that every count- State University, Wichita, KS ∈ n 67260‑0033, USA able union of closed complete pluripolar sets in P is the convergence set of some Full list of author information divergent series f. The (affine) convergence sets of formal power series with polynomial is available at the end of the article coefficients are also studied. The higher-dimensional analogs of the results of Sathaye (J Reine Angew Math 283:86–98, 1976), Lelong (Proc Am Math Soc 2:11–19, 1951), Lev- enberg and Molzon (Math Z 197:411–420, 1988), and of Ribón (Ann Scuola Norm Sup Pisa Cl Sci (5) 3:657–680 2004) are obtained. Mathematics Subject Classification: Primary: 32A05, 30C85, 40A05 Keywords: Formal power series, Convergence sets, Pluripolar sets, Projective hulls 1 Background A formal power series f (x0, x1, ..., xn) with coefficients inC is said to be convergent n 1 if it converges absolutely in a neighborhood of the origin in C + . A classical result of Hartogs (see [5]) states that a series f converges if and only if fz(t) f (z0t, z1t, ..., znt) n 1 := z C + converges, as a series in t, for all ∈ . 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 n 1 z C + for which fz converges. Since fz(t) converges if and only if fw(t) converges ∈ n 1 for all w C + on the affine line through z, ignoring the trivial case z 0, the set of ∈ n= directions along which f converges can be identified with a subset ofP . The conver- gence set Conv(f ) of a divergent power series f is defined to be the set of all directions n 1 n 1 n ξ P fz(t) z π − (ξ) π C + 0 P ∈ such that is convergent for some ∈ , where : \{ }→ is n 1 the natural projection. For the case = , Lelong [9] proved that the convergence set of a divergent series f (x1, x2) is an Fσ polar set (i.e. a countable union of closed sets 1 1 of vanishing logarithmic capacity) in P , and moreover, every Fσ polar subset of P is contained in the convergence set of a divergent series f (x1, x2). The optimal result was later obtained by Sathaye (see [16]) who showed that the class of convergence sets of 1 divergent power series f (x1, x2) is precisely the class of Fσ polar sets in P . To study n the collection Conv(P ) of convergence sets of divergent series in higher dimensions n n we consider the class PSHω(P ) of ω-plurisubharmonic functions on P with respect ω ddc log Z Pn Conv(Pn) to the form := | | on . We show that contains projective hulls of © 2015 Ma and Neelon. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 2 of 21 compact pluripolar sets and countable unions of projective varieties. We prove that each convergence set (of divergent power series) is a countable union of projective hulls of compact pluripolar sets. Our main result states that a countable union of closed com- n n plete pluripolar sets in P belongs to Conv(P ). 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]. We also consider convergence sets of a formal power series of the type ∞ j f (t, x) Pj(x)t C x1, x2, ..., xn t , = ∈ [ ][[ ]] j 1 = deg Pj j Conva(f ) where ≤ . The affine convergence set of a divergent power series f(t, x) x Cn is defined to be the set of all ∈ for which f(t, x) is convergent as a series in t. We prove that a countable union of closed complete pluripolar sets is an affine convergence set. 2 Pluripolar sets in Cn C x C x1, ..., xn Let [[ ]] := [[ ]] denote the ring of formal power series with complex coefficients in n indeterminates x1, ..., xn. Let C x be the ring of all power series { } n f (x) C x that are absolutely convergent in a neighborhood of the origin in C . ∈ [[n ]] Let H(C ) denote the set of all homogeneous polynomials (including the zero poly- H n nomial) in x1, ..., xn with complex coefficients. Fork 0 let k (C ) denote set of n ≥ n p H(C ) p 0 Hk (C ) C ∈ such that = or p is homogeneous of degree k. So each is a -vec- H n H n tor space and (C ) k∞ 0 k (C ). =∪ = n k 0, Pk (C ) For an integer ≥ let denote the set of polynomials of degree at most k in x1, ..., xn with complex coefficients. For convenience the zero polynomial is considered P n to have degree 1. So k (C ) contains 0 and it is a C-vector space for k 0. In particular n − ≥ P0(C ) C = . 1 Cn n 1 A Borel subset E of is said to be pluripolar (polar when = ) if for each point x E there is a plurisubharmonic function u, u , defined in a connected neigh- ∈ n �≡ −∞ borhood U of x in C such that u on E U. A set E is said to be globally pluripo- = −∞ ∩ n lar if there is a nonconstant plurisubharmonic function u defined onC such that E y u(y) ⊂{ : = −∞}. A theorem of Josefson (see [7]) states that E is pluripolar if and only if E is globally pluripolar. n A set E C is said to be a complete pluripolar set if there is a non-constant ⊂ n plurisubharmonic function u defined onC such that E y u(y) . So the set 2 ={ : = −∞} (0, x2) C x2 < 1 { ∈ :| | } and its closure are pluripolar, but they are not complete pluripo- lar sets. A countable union of pluripolar sets is pluripolar. So the set of rationals in the interval [0, 1] is polar. It is not a complete polar set, because each complete pluripolar set is Gδ. In C each Gδ polar set is a complete polar set, which is Deny’s Theorem (see [3]). n A subset E of a domain D in C is said to be a complete pluripolar set in D if there is a E u non-constant plurisubharmonic function u defined on D such that ={ = −∞}. 1 All sets considered in this paper are assumed to be Borel. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 3 of 21 Proposition 2.1 Let w be a nonconstant plurisubharmonic function defined on a Stein E w manifold , and let ={ = −∞}. Let v be a continuous, non-negative, plurisubhar- monic exhaustion function of . Then there is a plurisubharmonic function u on such u v E u that ≤ on and ={ = −∞}. j Vj x � v(x)<2 j N Mj Proof Let ={ ∈ : } for ∈ . Choose an increasing sequence { } of positive numbers such that limj Mj and Mj > supx V w(x) j. For each j, define →∞ =∞ ∈ j ∀ a function uj by 1 j max(Mj− w(x) 1), v(x) 2 ), if x Vj, uj(x) j − − ∈ = v(x) 2 , if x Vj. − �∈ Thenu j is plurisubharmonic on by the gluing theorem. On each compact subset of , j all but a finite number ofu j are negative. It follows that the sum u(x) j∞1 2− uj(x) := = is plurisubharmonic, since the sequence of the partial sums of the series is eventually decreasing. Since uj(x) v(x) for each j, we see that u(x) v(x). ≤ ≤ j x E w(x) uj(x) v(x) 2 Suppose that ∈ . Then = −∞, and = − for each j. Thus u(x) = −∞. x E x Vm Now suppose that ∈ \ . There is an m such that ∈ . Then m 1 − j ∞ j 1 u(x) 2− uj(x) 2− (M− w(x) 1) ≥ + j − j 1 jm = = m 1 − j ∞ j 1 2− uj(x) 2− ( M− w(x) 1) ≥ + − 1 | |− j 1 j 1 = = m 1 − j 1 2− uj(x) ( M− w(x) 1)> . = + − 1 | |− −∞ j 1 = E x u(x) Therefore, ={ : = −∞}. Remark We got the idea of the proof from Sadullaev (private discussion) and from Bedford and Taylor [2]. n 2 2 1/2 x C , x ( x1 xn ) For ∈ let | | := | | +···+| | . n Corollary 2.2 Let E be a complete pluripolar set in C . Then there is a plurisubhar- Cn u(x) (1/2) log(1 x 2) Cn E u monic function u on such that ≤ +| | on and ={ = −∞}. Remark A stronger version of the above corollary appeared in [2]. n Let Bn be the open unit ball in C . Corollary 2.3 Let E be a complete pluripolar set in Bn.
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