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
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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 j m = = 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. Then there is a plurisubhar- 2 Bn u(x) log(1 x ) Bn E u monic function u on such that ≤− −| | on and ={ = −∞}. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 4 of 21
Let
L(Cn) u PSH(Cn) sup (u(x) (1/2) log(1 x 2)) < ={ ∈ : x Cn − +| | ∞} ∈ P n denote the Lelong class of plurisubharmonic functions. For a polynomial P k (C ), n ∈ where k > 0, the function (1/k) log P(x) is a prototypical member of L(C ). | | n Corollary 2.2 implies that if E is a complete pluripolar set in C then there is a n u L(C ) such that E u . ∈ ={ = −∞}n n The pluripolar hull (see [11]) in C of a pluripolar set E in C is defined to be n E∗ x C u(x) , = ∩{ ∈ : = −∞} Cn where the intersection is taken over all plurisubharmonic functions u on that are −∞ on E. n n P Pk (C ) k > 0 C For a polynomial ∈ with and a subset K of we set P(x) 1/k P(x) k | | , P k,K sup P(x) k , P k P k,Cn . � � = 1 x 2 � � = x K � � � � =� � ∈ +| | n m P Pk (C ) P (x) P(x) Note that if ∈ and m is a positive integer, then � �km = � �k.
F Cn F x Cn 0 r 1 Definition 2.4 Let ⊂ , �= ∅, ∈ , and ≤ ≤ . Define
n τ(x, F, r) inf P k,F k N, P Pk (C ), P(x) k r, P k 1 = {� � : ∈ ∈ � � ≥ � � ≤ } and T(x, F) sup r 0 r 1, τ(x, F, r) 0 . = { : ≤ ≤ = } τ(x, , r) 0 T(x, ) 1 For the empty set, we put ∅ = and ∅ = . It follows directly from the E F τ(x, E, r) τ(x, F, r) T(x, E) T(x, F) definition that if ⊂ , then ≤ and ≥ .
u L(Cn) E u Lemma 2.5 Let ∈ . Suppose that the set := { = −∞} is closed. Then for each x Cn E K E ∈ \ , and each non-empty compact set ⊂ , we have
2 1/2 u(x) b T(x, K) (1 x )− e − , ≥ +| | where 1 b sup(u(z) log(1 z 2)). := z Cn − 2 +| | ∈ u(x) Proof Without loss of generality, we assume that b 0. Let g(x) e . Then 2 1/2 n n = = (1 x )− g(x) 1 x C x C E K E +| | ≤ for ∈ . Fix a ∈ \ and a non-empty compact set ⊂ . r > 0 r <(1 x 2) 1/2g(x) η η 1 ( η) − < √η. + (1) Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 5 of 21 Let 2 cn exp( 1/(1 y )), if y < 1, ω(y) − −| | | | ω(y) dy 1, = 0, if y 1, = | |≥ cn ω(t) dt 1 µ>0 gµ(y) g(y µz)ω(z) dz where is so chosen that = . For , let = + . L n Then gµ is C∞, positive, and gµ g as µ 0. We now show that log gµ (C ). Let f n ↓ n↓ ∈ y C a, ζ C ζ 0 log gµ Rf be a polynomial in ∈ and let ∈ with �= . Assume that ≤ on the ∂D a wζ w C, w 1 D a wζ w C, w 1 boundary := { + : ∈ | |= } of the disc := { + : ∈ | |≤ }, f R that is, gµ e on ∂D. Since u(y µz) f (y) is plurisubharmonic in y, its exponen- ≤| | f (y) + − f g(y µz) e gµ e tial + | − | is plurisubharmonic in y. Hence | − | is plurisubharmonic and 1 log gµ Rf log gµ therefore ≤ on D, that is, ≤ on D. This proves that is plurisubhar- y Cn monic. For ∈ , 2 1/2 gµ(y) (1 y µz ) ω(z) dz ≤ +| + | (1 ( y µ)2)1/2 ≤ + | |+ (1 µ)(1 y 2)1/2. (2) ≤ + +| | n log gµ L(C ) Therefore, ∈ . y K z < 1 y z u g(u) η g(y z)<η If ∈ , and if | | , then + �∈ { : ≥ }, and hence + . It follows that g (y) g(y z)ω(z) dz < ηω(z) dz η. By (2) we also have = / + = g (y) (1 )(1 y 2)1 2 y Cn ≤ + +| | for ∈ . As in ([15], p. 17) (we tried to find a more acces- φ C Cn sible reference without success), we define a function on × by 1 2 2 1/2 y0 ( g (y/y0)) − ( y0 y ) , if y0 0, φ (y0, y) | | + + | | +| | �= = y , if y0 0. | | = Then φ is continuous and plurisubharmonic. Moreover, it satisfiesφ (cw) c φ (w) n 1 =| | c C w C + ψ for ∈ and ∈ . We then define by 1 2 1/2 ψ (y) φ (1, y) ( g (y)) − (1 y ) . = = + + +| | n ψ C∞ log ψ L(C ) Then is and ∈ by [15, Prop. 2.7]. By [15, Prop. 2.10], 1/deg h φ (y , y) sup h(y , y) , 0 = {| 0 | } n 1 where the supremum is taken over all homogeneous polynomials h of + variables 1/deg h n 2 h(z0, z) φ (z0, z) (z0, z) C C ψ (z)/ 1 z such that | | ≤ ∀ ∈ × . Since +| | extends to n a continuous function on P , it follows that 1/k n 1/k n ψ (x) sup P(x) k N, P Pk (C ), P(y) ψ (y) y C . (3) = {| | : ∈ ∈ | | ≤ ∀ ∈ } y Cn For all ∈ , 1 2 1/2 ψ (y) ( g (y)) − (1 y ) = + + +| | 2 1/2 1 2 1/2 ( (1 )(1 y ) ) − (1 y ) ≤ + + +| | + +| | <(1 3 )(1 y 2)1/2. + +| | Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 6 of 21 y K If ∈ , then 1 2 1/2 ψ (y) ( g (y)) − (1 y ) = + + +| | 1 2 1/2 ( η) − (1 y ) ≤ + + +| | < √η (1 y 2)1/2 + +| | (√η )(1 y 2)1/2. ≤ + +| | n 1/k n P Pk (C ) P(z) ψ (z) z C If ∈ and if | | ≤ ∀ ∈ , then P k 1 3 and P k,K √η . (4) � � ≤ + � � ≤ + For sufficiently small , 1 2 1/2 2 1/2 ( g (x)) − (1 x ) >(1 3 )r(1 x ) , + + +| | + +| | since as approaches 0, the difference of the left side minus the right side tends to g(x) r(1 x 2)1/2 > 0 − +| | . It follows that for sufficiently small , 2 1/2 ψ (x)>(1 3 )r(1 x ) . (5) + +| | τ(x, K, r) (1 3 ) 1(√η ) 0 By (2), (3) and (4), we have ≤ + − + . Letting → , and then η 0 τ(x, K, r) 0 r < g(x)(1 x 2) 1/2 → , yields that = . Since this holds for every +| | − , it fol- T(x, K) g(x)(1 x 2) 1/2 lows that ≥ +| | − . Remark Some comments about (2) may be in order. By [8], Theorem 5.1.6 (iii)] 1/j ψ (lim sup Pj )∗ = j | | →∞ n Pj C deg Pj j w∗ for some sequence { } of polynomials on such that ≤ . Here denotes the upper semicontinuous regularization of w. The stronger equality (2) depends on the fact 2 n that ψ (z)/ 1 z extends to a continuous function on P . +| | n n n Definition 2.6 A subset E of C is said to have Property J (in C ) if for each x C E ∈ \n rx T(x, K E) rx C there is a positive number such that ∩ ≥ for each compact subset K of . T(x, K E) rx kj The inequality ∩ ≥ means that there is a sequence { } of positive integers, Pj degPj kj and a sequence { } of polynomials, with ≤ , such that Pj(x) k rx, Pj k 1, lim Pj k , K E 0. � � j ≥ � � j ≤ j � � j ∩ = (6) →∞ x E K y Cn y x 1 Suppose that E has Property J and �∈ . Let ={ ∈ :| |≤| |+ }. By (5), Pj(x) kj > Pj kj,K E for sufficiently large j, which implies that x does not belong to the � � � � ∩ E E E closure of E. It follows that \ =∅, and E is closed. Therefore, each set that has Prop- erty J must be closed. n Theorem 2.7 A subset of C has Property J if and only if it is a closed complete pluripo- lar set. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 7 of 21 n Proof Suppose that E is a closed complete pluripolar set in C . By Corollary 2.2, u L(Cn) E u there is a ∈ with ={ = −∞}. Without loss of generality, we assume that sup(u(z) 1 log(1 z 2)) 0 x Cn E − 2 +| | = . For each point ∈ \ and each compact set K, one has, by Lemma 2.5, that 2 1/2 u(x) T(x, K E) (1 x )− e . ∩ ≥ +| | Thus E has Property J. n Conversely, suppose that E C has Property J. Then E is closed. Set n ⊂ n Ej E z C z j x C E rx = ∩{ ∈ :| |≤ }. Let ∈ \ . Then there is a positive number such that T(x, Ej) rx kj ≥ for each positive integer j. Thus there is a sequence{ } of positive integers, Pj degPj kj and a sequence { } of polynomials, with ≤ , such that j Pj(x) rx, Pj 1, Pj exp( 2 ). � �kj ≥ � �kj ≤ � �kj, Ej ≤ − Let ∞ 1 ∞ j 1/kj 2 j u(z) 2− log Pj(z) log(1 z ) 2− log Pj(z) . := | | = 2 +| | + � �kj j 1 j 1 = = 2 n Then u is plurisubharmonic with u(z) (1/2) log(1 z ) on C , and 2 ≤ +| | u(x) log rx (1/2) log(1 x ) y E ≥ + +| | . Let ∈ . Then there is a positive integer m such that y Ej j m ∈ for ≥ , hence 1 2 ∞ j u(y) log(1 y ) 2− log Pj(y) = 2 +| | + � �kj j 1 = 1 2 ∞ j j log(1 y ) 2− ( 2 ) . ≤ 2 +| | + − = −∞ j m = u It follows that = −∞ on E. Thus x does not belong to the pluripolar hull of E. There- fore, the pluripolar hull of E is E. A Theorem of Zeriahi [17] states that if a pluripolar set F is both Fσ and Gδ, and if the pluripolar hull of F equals F, then F is a complete plurip- olar set. Since E, being a closed set, is both Gδ and Fσ, it follows that E is a complete pluripolar set. 3 Pluripolar sets in Pn n 1 n Let π C + 0 P denote the standard projection mapping that maps a : \{ }→ n 1 point z (z0, z1, ..., zn) C + to its corresponding homogeneous coordinates = ∈ n n 1 π(z) z z0 z1 ... zn P . Suppose that z (z0, z1, ..., zn) C + 0 and =[ ]=[ : : : n]∈ = ∈ \{ } Z Z0 Z1 Zn P Z π(z) =[ : :···: ]∈ . Then = if and only if Z Z Zn z z ... zn , [ 0 : 1 :···: ]=[ 0 : 1 : : ] or, equivalently, zjZ z Zj, for j, k 0, ..., n. k = k = Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 8 of 21 H n 1 n n 1 p k (C + ) k > 0 Z z P z C + 0 Suppose that ∈ with and that =[ ]∈ , where ∈ \{ }. Set p(Z) 1/k p(z) 1/k p(Z) | | | | . (7) � � := Z = z | | | | n p 0 p(Z) 0 p H0(C ) When ≡ , let � �= , which is consistent with (6). For ∈ , let p(Z) p(Z) . Note that p(Z) is independent of the choice of the representative z � �=| | n n P m, k > 0 p Hk (C ) and is a well-defined function on . Furthermore, if and ∈ , then pm(Z) p(Z) K Pn � �=� �. For a set ⊂ , put p K sup p(Z) , p p Pn . � � = Z K� � � �=� � ∈ n Definition 3.1 The projective hull Kˆ of a compact set K P is the set of all points n ⊂ Z P C CZ > 0 ∈ for which there exists a constant = such that p(Z) C p K (8) � �≤ � � n 1 n p H(C + ) K P for all homogeneous polynomials ∈ . A compact set ⊂ is said to be pro- K K jectively convex if ˆ = . k 1 H 0(Pn, O(k)) Since, for ≥ , the set of global holomorphic sections of the line bun- O H n 1 dle (k) is canonically identified with the set k (C + ) of homogeneous polynomials of degree k, it follows that the above definition of projective hulls is equivalent to that in [6, p. 607]. n n 1 K p 0 P p H(C + ) Consider an algebraic variety := { = }⊂ , where ∈ . Assume that Z K Z K �∈ . Then (7) does not hold since the right side equals 0, which implies that �∈ ˆ . K K Thus, ˆ \ is empty, and hence K is projectively convex. Therefore, each algebraic vari- ety is projectively convex. In particular, each finite set is projectively convex. n The complement of a hyperplane inP is called an affine open set. An affine open set n n in P is biholomorphically equivalent to C . The sets n Uj Z Z0 Z1 Zn P Zj 0 , j 0, ..., n, (9) := { =[ : :···: ]∈ : �= } = n are canonical affine open sets inP . n Definition 3.2 A subset F of P is said to be a pluripolar set if for each Z F there is a n ∈ neighborhood V of Z in P and a nonconstant plurisubharmonic function u defined onV V F Pn such that u is identically −∞ on ∩ . A subset E of is said to be a complete pluripo- Pn Pn U E lar set in if for each affine open set U in the set ∩ is a complete pluripolar set in U. Pn K Pn We emphasize that a compact set K in is pluripolar if and only if ˆ �= (see [6, Corollary 4.4]). In the above definition, the pluripolar sets and the complete pluripolar sets are defined “locally”, because there are no globally defined nonconstant plurisubharmonic func- n tions on P . It is desirable to have an equivalent definition of (complete) pluripolar sets in terms of some kind of substitutes of plurisubharmonic functions that are glob- n ally defined onP . We could use the ω-plurisubharmonic functions described below to Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 9 of 21 define pluripolar sets and complete pluripolar sets. Definition 3.2 is to emphasize that the notion of (complete) pluripolar sets is independent of any differential forms. We fix a Kähler form c 2 2 ω dd log Z i∂∂ log( Z Zn ) := | |= | 0| +···+| | n c 1 n on P , where d i(∂ ∂). Note that (2π)− ω is the Fubini-Study form on P . = − n An upper semicontinuous function u from an open subset of P to R is said c ∪ {−∞}n to be ω-plurisubharmonic if dd u ω 0 (see, e.g., [4]). Let PSHω(P ) denote + ≥ n the family of ω-plurisubharmonic functions on P . For a homogeneous poly- n Z log p(Z) PSHω(P ) nomial p, the function �→ � � is a prototypical function in . Suppose that ℓ(Z) a0Z0 anZn is a linear form and Q is an open subset of n := +···+ Uℓ Z P ℓ(Z) 0 ω := { ∈ : �= }. Then a function u on Q is -plurisubharmonic if and only if the function u(Z) log( Z / ℓ(Z) ) is plurisubharmonic. + | | | | n n There is a one to one correspondence betweenPSH ω(P ) and the Lelong class L(C ). n This can be seen by identifyingC with the affine open setU 0: n n n C 1 ξ1 ξ2 ... ξn P (ξ1, ..., ξn) C U0. (10) ≃ [ : : : : ]∈ : ∈ = n ϕ PSHω(P ), Given ∈ the function 1 2 ϕ(z1, ..., zn) log(1 z ) ϕ(1 z zn) ˆ := 2 +| | + : 1 :···: n n n L(C ) ϕ ϕ PSHω(P ) L(C ) belongs to , and the map �→ ˆ is a bijection from onto (see [4]). The following proposition can be found in [4] or [6, Theorem 4.3]. n Proposition 3.3 Let E be a subset of P . Then E is pluripolar if and only if there is a n u PSHω(P ) u E u function ∈ , �≡ −∞, such that ⊂{ = −∞}. n 1 n Lemma 3.4 Suppose that n 2 and H ∼ P − is a hyperplane in P . Let u be an ≥ = n ω-plurisubharmonic function on H. Then there is an ω-plurisubharmonic function v on P such that Z Pn v(Z) Z H u(Z) . (11) { ∈ : = −∞} = { ∈ : = −∞} u 0 Proof Without loss of generality we assume that ≤ and H Z Z Zn Z 0 O 1 0 0 ={ =[ 0 :···: ]: 0 = }. Let =[ : :···: ]. The function 1 Z 2 Z 2 ( ) ( ) 1 n w Z u 0 Z1 Zn log 2| | +···+|2 | 2 := : :···: + 2 Z Z Zn | 0| +| 1| +···+| | n ω P O w H u is an -plurisubharmonic function on \ such that | = . Let n max(w(Z),1 log( Z0 / Z )), if Z P O, v(Z) + | | | | ∈ \ = 1, if Z O. = 1 log( Z / Z ) > 0 w U O Since the function + | 0| | | is ≥ on \ for some neighborhood U of O, v(Z) 1 log( Z / Z ) U O v(Z) 1 log( Z / Z ) we see that = + | 0| | | on \ , hence = + | 0| | | on U. It Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 10 of 21 n follows that v PSHω(P ). Since v w on H, and since v > on U, it follows that ∈ = −∞ (10) holds. n n Proposition 3.5 Let E P . Then E is a complete pluripolar set in P if and only if ⊂ n u PSHω(P ) E Z u(Z) there is a non-constant ∈ such that ={ : = −∞}. n Proof Suppose that there is a u PSHω(P ) such that E u . Let U be an affine n ∈ ={ = −∞} H P U Z �(Z) 0 �(Z) a0Z0 a1Z1 anZn open set, and let = \ ={ : = }, where := + +···+ v(Z) u(Z) log( Z / �(Z) ) is a linear form. Then := + | | | | is plurisubharmonic on U and U E Z U u(Z) Z U v(Z) . ∩ ={ ∈ : = −∞} = { ∈ : = −∞} It follows that U E is a complete pluripolar set in U for each affine open set U. There- ∩ n fore, E is a complete pluripolar set in P . n Conversely, suppose that E is a complete pluripolar set in P . For j 0, ..., n, let n = Hj Zj 0 Uj P Hj E Uj Uj ={ = } and = \ . Since ∩ is a complete pluripolar set in , there hj Uj E Uj Z Uj hj(Z) is a plurisubharmonic function on with ∩ ={ ∈ : = −∞} and hj(Z) log( Z / Zj ) hj(Z) log( Z / Zj ) ≤ | | | | , by Corollary 2.2. The function − | | | | is a non-pos- itive ω-plurisubharmonic function on Uj, which extends uniquely to an ω-plurisubhar- n monic function wj on P . w max wj Let = . For each j, Z Uj w(Z) Z Uj wj(Z) E Uj. { ∈ : = −∞} ⊂ { ∈ : = −∞} = ∩ w E U Un It follows that { = −∞} ⊂ . Let = 0 ∩···∩ . Then Z � wj(Z) � Z Uj wj(Z) � (E Uj) E �, { ∈ : = −∞} = ∩{ ∈ : = −∞} = ∩ ∩ = ∩ for each j, hence w � j( wj �) E �. { = −∞} ∩ =∩ { = −∞} ∩ = ∩ To summarize, we have w E and w E . { = −∞} ⊂ n { = −∞} ∩ = ∩ u PSHω(P ) E u To prove that there is a ∈ with ={ = −∞}, we proceed by induction on n. Suppose that n 1. If 0 1 E, let v0 log( Z0 / Z ); if 0 1 E, let v0 0. 1 = [ : ]∈ = | | | | [ : ] �∈ = 1 Then v0 PSHω(P ) and v0 E H0. We similarly definev 1 PSHω(P ) ∈ { = −∞} = ∩ ∈ 1 v1 E H1 u (w v0 v1)/3 PSHω(P ) so that { = −∞} = ∩ . Then := + + belongs to and u E n 1 { = −∞} = . The statement holds for = . n 2 n 1 Hj E Suppose that ≥ and that the statement holds for − . For each j, either ⊂ , Hj E Hj H E v log( Z / Z ) or ∩ is a complete pluripolar set in . If 0 ⊂ , let 0 = | 0| | | . If H0 E, then, by the induction hypothesis, there is a u0 PSHω(H0) with �⊂ ∈ n Z H0 u0(Z) E H0, and hence, by Lemma 3.4, there is a v0 PSHω(P ) { ∈ : n = −∞} = ∩ ∈ n with Z P v0(Z) E H0. Either case, we have v0 PSHω(P ) { ∈ : = −∞} = ∩ ∈ n v0 E H0 vj vj PSHω(P ) and { = −∞} = ∩ . We similarly obtain so that ∈ and vj E Hj, for each j. Thenu (w v0 vn)/(n 2) belongs to { = −∞}n = ∩ := + +···+ + PSHω(P ) u E and { = −∞} = , which completes the proof. Propositions 3.3 and 3.5 can be considered equivalent definitions of pluripolar sets and complete pluripolar sets respectively. Since Definition 3.2 is local, the union of a Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 11 of 21 n countable collection of pluripolar sets in P is pluripolar for the same reason that the corresponding statement is true in affine space. It is a consequence of Propositions 3.3 n n and 3.5 that each pluripolar set in P is contained in a complete pluripolar set in P . n n Let E be a pluripolar set in P . The intersection of all complete pluripolar sets inP n that contain E is called the pluripolar hull (see [11]) of E (in P ), and is denoted by E∗. n P K K ∗ Proposition 3.6 Let K be a compact pluripolar set in . Then ˆ ⊂ . Proof Let n LK (Z) sup ϕ(Z) ϕ PSHω(P ) and ϕ K 0 = { : ∈ | ≤ } and n 1 �K (Z) sup log p(Z) p H(C + ), p K 1 . = { � �: ∈ � � ≤ } Then� K (Z) LK (Z), by [6, Preposition 4.2]. =c n X K ∗ K ∗ u PSHω(P ) u(X)> Let ∈ , the complement of . Then there is a ∈ with −∞ u K LK (X) �K (X) and | ≡ −∞. Thus =∞. It follows that =∞, which implies that X K c K K c K c K K ∈ ˆ , the complement of ˆ . Therefore, ∗ ⊂ ˆ , which is equivalent to ˆ ⊂ ∗. n Corollary 3.7 Each compact complete pluripolar set in P is projectively convex. E 1 z ez P2 Remark The converse to Corollary 3.7 is false. The set := {[ : : ]∈ : z C, z 1 P2 ∈ | |≤ } is projectively convex in (see [6]), but it is not a complete pluripolar set. n Proposition 3.8 Let U be an affine open set in P , and let E be a compact complete n pluripolar set in U. Then E is a complete pluripolar set in P . U Z 0 Proof Without loss of generality, we assume that ={ 0 �= } and E Z U Z/Z < √2 ⊂{ ∈ :| 0| }. By the proof of ([10, Lemma 5.4) there is a number a > 1 and a plurisubharmonic function u defined on U such that 2 2 E X U u(X) u(Z) log( Z Zn / Z ) ={ ∈ : = −∞} and = | 1| +···+| | | 0| on the set 2 2 Z U Z1 Zn / Z0 a . Let { ∈ : | | +···+| | | |≥ } u(Z) log( Z / Z ), if Z U, v(Z) − | | | 0| ∈ = 0, if Z U. �∈ n n v PSHω(P ) E Z P v(Z) Then ∈ and ={ ∈ : = −∞}. Therefore, E is a complete pluripolar n set in P . 4 Convergence sets in Pn n 1 f (x) C x C x0, x1, ..., xn z C + For a given ∈ [[ ]] = [[ ]], we are interested in the set of all ∈ fz(t) f (z0t, z1t, ..., znt) C t C 0 , for which the restriction := ∈ { }. Since for ∈ \{ } the series fz(t) and f z either both converge or both diverge, it is more appropriate to consider the set of affine lines along whichf converges. Thus the set of affine lines Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 12 of 21 n along which f converges can be identified as a subset of the projective spaceP . Since n 1 f C x fz(t) C t , z C + a ∈ [[ ]] converges if and only if ∈ { } ∀ ∈ , our focus will be on the divergent power series. The convergence set of a power series f C x , denoted by Conv(f ), is the set of all n ∈ [[ ]] 1 Z P fz(t) C t z π − (Z) ∈ such that ∈ { } for some (and hence all) ∈ . Then f converges if Conv(f ) Pn and only if = . n n Definition 4.1 A subset E of P is said to be a convergence set (in P ) if E Conv(f ) n = for some divergent power series f. Let Conv(P ) denote the collection of all convergence n sets in P : n Conv(P ) Conv(f ) f C x0, x1, ..., xn , f diverges . := { : ∈ [[ ]] } Convergence sets were first studied in [1]. f C x0, x1, ..., xn Consider a divergent series ∈ [[ ]]. Since ∞ j n 1 fz(t) f (z0t, z1t, ..., znt) pj(z)t , pj Hj(C + ), := = ∈ j 1 = we see that n Conv(f ) Z P sup pj(Z) < . ={ ∈ : j � � ∞} In fact, we have the following lemma (see [16]). E �Pn E Conv(Pn) Lemma 4.2 Suppose that . Then ∈ if and only if there exists a count- F H n 1 able family of non-constant homogeneous polynomials in (C + ) such that E Z Pn sup p(Z) < . ={ ∈ : p F� � ∞} (12) ∈ F Note that there is no need to require the degrees of the polynomials in to form a n strictly increasing sequence. Suppose that E �P, and that there exists a countable fam- F H n 1 ily (C + ) of non-constant homogeneous polynomials such that (11) holds. Let ⊂ kj pj F pj qj p { } be an enumeration of . Raise to suitable powers to obtain := j so that the degqj pj(Z) qj(Z) sequence { } is strictly increasing. Since � �=� �, we see that n E Z P sup qj(Z) < . ={ ∈ : j � � ∞} E Conv(g) g(x) ∞ qj(x) It follows that = , where = j 1 . = E Conv(Pn) F Conv(Pn) E F Conv(Pn) Proposition 4.3 If ∈ and ∈ then ∩ ∈ . Proof Suppose that E and F are convergence sets. By Lemma 4.2, there are countable E F families and of homogeneous polynomials such that Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 13 of 21 E Z Pn sup p(Z) < , F Z Pn sup p(Z) < . ={ ∈ : p E � � ∞} ={ ∈ : p F� � ∞} ∈ ∈ It follows that E F Z Pn sup p(Z) < . ∩ ={ ∈ : p E F� � ∞} ∈ ∪ E F Conv(Pn) Therefore, ∩ ∈ . We do not know whether the union of two convergence sets is necessarily a conver- gence set. n Proposition 4.4 Suppose that K is a compact pluripolar subset of P . Then K Conv(Pn) Pn ˆ ∈ . In particular, each projectively convex compact pluripolar set in is a convergence set. Proof The proposition is proved by following the approach of [10, Theorem 5.6]. Let n 1 F p p Hj(C + ) with j 1, p K 1 . ={ : ∈ ≥ � � ≤ } Then K Z Pn sup p(Z) < . ˆ ={ ∈ : p F� � ∞} ∈ K Pn E Since K is a compact pluripolar set, ˆ �= (see [6, Corollary 4.4]). Let be the set of F Q iQ polynomials in whose coefficients belong to + , the set of complex numbers Q iQ C whose real and imaginary parts are rational numbers. Since + is dense in , we see that K Z Pn sup p(Z) < . ˆ ={ ∈ : p E � � ∞} ∈ E n Since is countable, the set Kˆ is a convergence set in P by Lemma 4.2. Remark Proposition 4.4 is motivated by Theorem 5.6 in [10], which states that if U is n an affine open set inP and if K is a compact complete pluripolar set in U then K is the n intersection of U and a convergence set in P . 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) Kˆ may be non- compact, (b) Kˆ is not necessarily a complete pluripolar set, and (c) Kˆ does not have to lie in an affine open set. n Corollary 4.5 Each algebraic variety in P is a convergence set. In particular, each finite set is a convergence set. n Theorem 4.6 Let E be a convergence set in P . Then there exists an ascending sequence Kj E Kj Kj { } of compact pluripolar sets such that =∪ =∪ˆ . In particular, E is a pluripolar Fσ set. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 14 of 21 H n 1 Proof Put E Conv(f ) and f (z) m∞ 1 fm(z), where fm m(C + ). Then = = = ∈ E jKj =∪ where n Kj Z P fm(Z) j , m . ={ ∈ :� �≤ ∀ } X Kj ℓ Fix a j and suppose that ∈ ˆ . There is a positive integer such that p(X) ℓ p K � �≤ � � j H n 1 for each p (C + ), and in particular for p fm. Thus X Kℓj E. It follows that ∈ = n ∈ n ⊂ Kˆ j E for each j and E jKˆ j. For a compact set K P , Kˆ P if and only if K and ⊂ =∪ ⊂ n �= Kˆ are pluripolar (see, e.g., [6, Corollary 4.4]). Since E P , we see that Kj and Kˆ j are �= pluripolar for each j. n k Lemma 4.7 Let Kj be a sequence of compact pluripolar sets in P such that j 1Kj is { } ∪ = projectively convex for each k, and let E Kj. Let Uj be a sequence of open sets such k =∪ { } n that Uk j 1Kj and j∞k Uj E for each k. Then E Conv(P ). ⊃∪= ∩ = ⊂ ∈ j n Proof Let Ej i 1Ki and Ŵj P Uj for j 1, 2, .... Fix a j and let X Ŵj. Since X =∪= = \ = ∈ does not belong to Ej, a projectively convex set, there exists a homogeneous polynomial pX such that pX (X) > j and pX E 1. � � � � j ≤ n Z P pX (Z) > j X Ŵj Ŵj Since the sets { ∈ :� � }, where ∈ , form an open cover of , there exist X , ..., Xℓ Ŵj 1 j ∈ such that ℓj i 1 Z pXi (Z) > j Ŵj. ∪ = { :� � }⊃ pji pX Put = i. Then pji E 1, � � j ≤ and max pji(X) > j, for X Ŵj. 1 i ℓ � � ∈ (13) ≤ ≤ j X E X E pji(X) 1 Suppose that ∈ . Then there is a positive integer k such that ∈ k. Thus� �≤ j k pji j < k,1 i ℓj for ≥ . Since the set { : ≤ ≤ } is finite, we see that sup pji(X) < . j,i � � ∞ (14) n X P sup pji(X) k Conversely, suppose that ∈ and (13) holds. Then j,i� �≤ for some posi- tive integer k. It follows from (12) that X Ŵj for j k. Thus X j∞k Uj E. �∈ ≥ ∈∩= ⊂ X E E Conv(Pn) To summarize, ∈ if and only if (13) holds. By Lemma 4.2, ∈ . Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 15 of 21 Proposition 4.8 Let Kj be an ascending sequence of projectively convex, compact, { } n E jKj Gδ E Conv(P ) pluripolar sets such that := ∪ is . Then ∈ . Proof Since E is Gδ, there is a descending sequence Uj of open sets such that { n } E Kj Uj E Conv(P ) =∪ =∩ . It follows from Lemma 4.7 that ∈ . Proposition 4.9 Let Kj be a sequence of pairwise disjoint compact pluripolar sets in n { } k P such that for each positive integer k the set j 1Kj is projectively convex. Then K Kj n ∪ = := ∪ is a convergence set in P . n Proof Let d denote a distance function on P that induces the standard topology. Let r (1/2) min(1/k, min d(Ki, Kj) 1 i < j k 1 ). k = { : ≤ ≤ + } k Let Uk be the rk-neighborhood of j 1Kj. ∪ = Suppose that m is a positive integer and that X k∞ mUk. We claim that ∈∩ = m d(X, j 1Kj) ℓ ℓ m X Um We will prove (14 ) by induction on . If = , then (14) holds because ∈ . Sup- ℓ>m ℓ ℓ 1 X Uℓ pose that and (14) holds with replaced by − . Since ∈ , we have ℓ d(X, j 1Kj) On the other hand, , ℓ m , ℓ , m 2 , d(X j m 1Kj) d( j 1Kj j m 1Kj) d(X j 1Kj) rℓ 1 rℓ 1 rℓ 1 rℓ ∪ = + ≥ ∪ = ∪ = + − ∪ = ≥ − − − = − ≥ hence ℓ d(X, j m 1Kj) rℓ. (17) ∪ = + ≥ m The inequalities (15) and (16) imply that d(X, j 1Kj) Let ϕ(z) be a non-polynomial entire function defined on the complex plane. ForS C, 2 ⊂ let S˜ be the subset of P defined by S 1 z ϕ(z) P2 z S . ˜ = {[ : : ]∈ : ∈ } By Theorem 9.2 in [6], which depends on a deep theorem in [14], for a compact set K C K C K ⊂ , the set ˜ is projectively convex if and only if \ is connected. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 16 of 21 Example 4.11 The setC ˜ is a convergence set. To see this, write C˜ ˜ j, where 2 =∪ j z C z j C Gδ P j ={ ∈ :| |≤ }. Since ˜ is a set in and since each ˜ is projectively convex, C Conv(P2) we see that ˜ ∈ by Proposition 4.8. Ŵ C Ŵ Conv(P2) Example 4.12 Let be the unit circle in . Then ˜ ∈ . To see this, write Ŵ Sj ˜ =∪˜ , where it Sj e 0 t 2π 1/j . ={ : ≤ ≤ − } Ŵ G S Ŵ Since ˜ is δ and since each ˜j is projectively convex, we see that ˜ is a convergence set by Proposition 4.8. Example 4.13 Let S be the subset of C obtained by removing an open triangle from a closed triangle. ThenS ˜ is a convergence set. This follows from an argument very similar to the previous example. Example 4.14 Let S be the subset of C obtained by removing a finite number of open triangles from a closed triangle. ThenS ˜ is a convergence set. This follows from the previ- ous example and Proposition 4.3. Let be a closed triangle in C. The “open middle triangle” of is the open triangle whose vertices are the midpoints of the sides of . Let V1 be the open middle triangle of V V , V , V . The set \ 1 is the union of three congruent closed triangles. Let 2 3 4 denote V the open middle triangles of the three closed triangles whose union is \ 1. Similarly, let V5, ..., V13 denote the open middle triangles of the nine closed triangles whose union 4 is j 1 Vj. Continuing in this way, we obtain a sequence Vj of open triangles. The \∪= { } k set E j∞1 Vj is called Sierpiński’s triangle. Let Ek j 1 Vj for k 1, 2, .... := \∪= 2 = \∪= = By Example 4.14, each Ek is a convergence set in P . Note that E k∞ 1Ek. ˜ ˜ =∩ = ˜ Proposition 4.15 The set E˜ is not a convergence set. E Conv(f ) f pm pm Proof Seeking for a contradiction, suppose that ˜ = and = , where is a homogeneous polynomial of degree m in three variables z0, z1, z2. Let 1/m Tk z C pm(1, z, ϕ(z)) k m . ={ ∈ :| | ≤ ∀ } Tk C E Tk Then each is a closed subset of and =∪ . By the maximum modulus principle, C Tk \ is connected for each k. We observe that the set E has the following property: if S is a closed set in the topological space E with nonempty interior, then S contains the sides of a small triangle and hence C S is disconnected. It follows that each Tk has empty inte- \ rior, which contradicts the Baire category theorem. ( 2) Let E˜k be the sequence defined above. Note that each E˜k Conv P , but { } 2 ∈ Ek Conv(P ) ∩ ˜ �∈ . Fix a positive number M. Let S 1 0 0 , and for k 1, ..., n, 0 = {[ : :···: ]} = n 2 2 2 2 Sk X P X0 Xk 1 M Xk , Xk 1 Xn 0 . ={ ∈ :| | +···+| − | ≤ | | + =···= = } Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 17 of 21 Put n KM k 0Sk . (18) =∪ = n Then Km is an ascending sequence of closed sets with P m∞ 1Km. { } n =∪ = n Recall that the set of all hyperplanes in P is naturally isomorphic to P . The set of all n hyperplanes in P passing through a fixed point is a hyperplane in . Pn ξ Pn Lemma 4.16 If K is a closed subset of contained in an affine open set, and if ∈ , K ξ then ∪{ } is contained in an affine open set. R H H K S H � ξ H Proof Let ={ ∈ : ∩ = ∅} and ={ ∈ : ∈ }. Then R is a non-empty R S open set in and S is a hyperplane in . Thus \ is non-empty. This means that there H (K ξ ) K ξ is a hyperplane H with ∩ ∪{ } =∅. Therefore, ∪{ } is contained in the affine Pn H open set \ . n 1 If 1 ν n and if u0, ..., uν are linearly independent vectors in C + , let ≤ ≤ n span(u0, ..., uν) be the ν-dimensional linear space in P defined by ν 1 span(u0, ..., uν) π(c0u0 cνuν) (c0, ..., cν) C + 0 , := { +···+ : ∈ \{ }} n 1 n π C + 0 P where : \{ }→ is the standard projection. Lemma 4.17 For each M > 0, the set KM is contained in an affine open set. n 1 Proof Let e0, ..., en be the standard basis of C + , and let ε be a sufficiently small posi- tive number. Let vj ej εej 1 for j 0, ..., n 1. = + + = − Put Vj span(v0, ..., vj) for j 0, ..., n 1, and V Vn 1. Also, let = = − = − Wj span(e , ..., ej) = 0 . Note that V Wj Vj 1, for j 1, ..., n. ∩ ⊂ − = Since Sj Wj, it follows that V Sj Vj 1 for j 1. Since the vectors e0, v0, v1, ..., vn 1 ⊂ ∩ ⊂ − ≥ − are linearly independent we see that V S0 . For j 1, since Wj 1 Sj , and ∩ =∅ ≥ − ∩ =∅ since Vj 1 is close to Wj 1 for sufficiently smallε , we see that Vj 1 Sj . It follows that − − − ∩ =∅ n n V KM j 0(V Sj) j 1(Vj 1 Sj) . ∩ =∪= ∩ ⊂∪= − ∩ =∅ n KM P V Therefore is contained in the affine open set \ . Theorem 4.18 The union of a countable collection of closed complete pluripolar sets in n n P is a convergence set in P . n Em P E Em Proof Let { } be a sequence of closed complete pluripolar sets in and let =∪ . Without loss of generality, we assume that the sequence Em is ascending, since the { } n union of a finite number of closed complete pluripolar sets inP is a closed complete n P Km Km Km pluripolar set. Recall that =∪ , where { } is ascending and each is a compact E (Em Km) set contained in an affine open set. We have =∪ ∩ . Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 18 of 21 For each positive integer m, we shall construct a sequence hmk k∞ 1 of homogeneous { } = polynomials such that for all k, (i) hmk Km Em 1, � � ∩ ≤ h m (ii) � mk �≤ , and n n (iii) k∞ 1 X P hmk (X) > m/2 P Em. ∪ = { ∈ :� � }⊃ \ n Y P Em Fix a positive integer m. Let ∈ \ . By Lemmas 4.16 and 4.17, we see that (Km Em) Y is contained in an affine open set V. Since V Em is a (relatively) closed ∩ ∪{ } n ∩ V C V Em complete pluripolar set in ≈ , the set ∩ has Property J in V by Theorem 2.7. Hence there is a number r with 0 < r < 1 such that τ(Y , Km Em, r) 0, which means, ∩ = n 1 pj H(C + ) in terms of homogeneous coordinates, that there is a sequence { } in such that lim pj Km Em 0, and for all j, pj(Y ) r, pj 1. j � � ∩ = � �≥ � �≤ →∞ Choose a positive rational number β a/b < 1, where a, b are positive integers, such β = n 1 (r/m) > 1/2 p H(C + ) that . There is a homogeneous polynomial ∈ such that 1/β n p Km Em < m− , p(Y ) r, and p P 1. � � ∩ � �≥ � � ≤ 1 n 1 v degp y π − (Y ) q H(C + ) Let = and ∈ . Define ∈ by v 1 v q(x) (mx y/ y ) m y − (x y xny ) . = · | | =[ | | 0 0 +···+ n ] By the Cauchy–Schwarz inequality, we have q Pn m q(Y ) . � � = =� � a b a n 1 n h p q − h H(C + ) deg h bv X P Let = . Then ∈ and = . For each ∈ , h(X) p(X) β q(X) 1 β � �=� � � � − . Hence, 1 β h Pn m − m, � � ≤ ≤ 1 1 β β h Em Km < m− m − m− 1, � � ∩ = ≤ β 1 β β h(Y ) r m − (r/m) m > m/2. � �≥ = n 1 hY H(C + ) To summarize, there is an ∈ such that n hY P m, hY Em Km < 1, and hY (Y ) > m/2. � � ≤ � � ∩ � � Let UY X hY (X) > m/2 . ThenU Y is a neighborhood of Y. The open cover := {n :� �n } UY Y P Em P Em UY k 1, 2, ... { : ∈ \ } of \ contains a countable subcover { k : = }. Put h hY h mk = k. Then the sequence{ mk } satisfies (i), (ii) and (iii). X E m 2 X (Em Km ) Suppose that ∈ . Then there is an 0 ≥ such that ∈ 0 ∩ 0 . h (X) 1 m m h (X) m 1 m < m We have � mk �≤ for ≥ 0, and � mk �≤ 0 − for 0. Hence, supm,k hmk (X) m0 1 < . � �≤ −n ∞ n X P E X P Em sup hmk (X) m/2 Suppose that ∈ \ . For each m, ∈ \ , hence k � �≥ . Thus sup h (X) m,k � mk �=∞. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 19 of 21 Therefore, n E X P sup hmk (X) < . ={ ∈ : m,k � � ∞} E Conv(Pn) By Lemma 4.2, ∈ . n Corollary 4.19 The union of a countable collection of algebraic varieties in P is a con- n vergence set in P . � 1 z ϕ(z) z C, z 1 The converse of Theorem 4.18 is not true. The set := {[ : : ]: ∈ | |= } in Example 4.12 is a convergence set, but it is not a countable union of complete plurip- olar sets. Recall that ϕ is a non-polynomial entire function defined onC . Seeking for 2 Fj Fj P a contradiction, suppose that =∪ , where are complete pluripolar sets in . Let Q 1 z ϕ(z) z C . For each j, since Fj Q, and since Fj is a complete pluripo- := {[ : 2: ]: ∈ } P Fj Fj lar set in , it follows that is polar in Q. Therefore, =∪ is polar in Q, which is false. 5 Affine convergence sets j n f (t, x) ∞ Pj(x)t C x1, ..., xn t Let be the set of series = j 0 ∈ [ ][[ ]] such that P n = Pj k (C ). (Here we use the convention that the degree of the zero polynomial is 1; ∈ n n − 0 Pk (C ) f n Conva(f ) x C hence ∈ .) For ∈ , let be the set of ∈ for which f(t, x) con- verges as a series of a single indeterminate t: n Conva(f ) x C f (t, x) C t . := { ∈ : ∈ { }} n Conva(f ) C f C t, x1, ..., xn By Hartogs’ theorem, = if and only if ∈ { }, the set of conver- gent series in t, x1, ..., xn. n n Definition 5.1 A subset E of C is said to be an affine convergence set (inC ) if n E Conva(f ) for some divergent power series f n. Let Conva(C ) denote the collec- = n ∈ tion of all affine convergence sets inC : n Conva(C ) Conva(f ) f �n, f diverges . := { : ∈ } Affine convergence sets were studied in [12, 13]. x Conva(f ) Note that if ∈ for a divergent series f it does not follow that x Conva(f ) C, and therefore π(Conva(f )) in general is not a convergence set ∈n 1 ∀ ∈ in P − . In analogy with Lemma 4.2, we have the following lemma. n n Lemma 5.2 Suppose that E �C. Then E Conva(C ) if and only if there exists a ∈ n kj N Pj Pj Pk (C ) sequence { }⊂ and a sequence { } of polynomials with ∈ j for all j such that n E x C sup Pj(x) kj < . ={ ∈ : j � � ∞} (19) n n n Let ι C P be defined byι( x1, ..., xn) 1 x1 xn . Then ι embeds C into n : → n =[ n: :···: ] n P C ι(C ) U0 Z P Z0 0 and identifies with the affine open set = := { ∈ : �= }. Define τ �n C x0, ..., xn : → [[ ]] by Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 20 of 21 ∞ j ∞ j τ Pj(x)t x Pj(x/x0). = 0 �j 0 �j 0 = = j n 1 pj(x0, x) x Pj(x/x0) Hj(C + ) j 0 Note that := 0 belongs to for ≥ . We also have Pj(x) j pj(ι(x)) j 1 � � =� � for ≥ . It follows from Lemmas 4.2 and 5.2 that ι(Conva(f )) U Conv(τ(f )). = 0 ∩ Consequently, we have the following proposition. n n Proposition 5.3 Suppose that E �C. Then E Conva(C ) if and only if n ∈ ι(E) U0 F F Conv(P ) = ∩ for some ∈ . n For an affine convergence set E in C , the set ι(E) may or may not be a convergence set n in P . ( x) 2 ι( ) Example 5.4 Let E x, e x C C . Then the set E equals the set C˜ in Exam- ={x : ∈ }⊂ 2 ple 4.11 when ϕ(x) e . It follows that ι(E) is a convergence set in P , and hence E is an = 2 affine convergence set inC . 2 Example 5.5 Let E (x,0) x C C . Thenι( E) U0 H, where ={ : ∈ }⊂ 2 = ∩ H Z0 Z1 Z2 Z2 0 . Since H is a convergence set in P by Corollary 4.5, we = {[ : : ]: = } 2 1 see that E is an affine convergence set inC . Since H is a copy of P , it follows from 2 Theorem 4.6 that for each convergence set Q in P , either H Q, or H Q is polar in H. 2 ⊂ ∩ ι(E) U0 H P Therefore, = ∩ is not a convergence set in . n For a compact subset K of C , let 1 Ka ι− (U ι(K)). ˆ = 0 ∩ Intuitively, Kˆ a is the projective hull of K minus the part at . Equivalently, Kˆ a is the set n ∞ of all x C for which there is a constant C Cx > 0 such that P(x) j C P j,K for all ∈ n = � � ≤ � � j N P Pj(C ) ∈ and all ∈ . n Theorem 5.6 Let E be an affine convergence set in C . Then there exists an ascending Kj E Kj Kj a sequence { } of compact pluripolar sets such that =∪ =∪ˆ , . In particular, E is a pluripolar Fσ set. Theorem 5.7 The union of a countable collection of closed complete pluripolar sets in n n C is an affine convergence set in C . n Corollary 5.8 The union of a countable collection of analytic varieties in C is an affine n convergence set in C . n Corollary 5.9 The finite sets and the countable sets in C are affine convergence sets. Ma and Neelon Complex Analysis and its Synergies (2015) 1:4 Page 21 of 21 The proofs of Theorems 5.6 and 5.7 are routine modifications of the respective proofs of Theorems 4.6 and 4.18. Note that Theorem 5.7 is not a consequence of Theorem 4.18, n since a complete pluripolar set in an affine open set inP in general does not extend to a n complete pluripolar set in P . Authors’ contributions DM and TSN discussed the topic and obtained the results and proofs. DM wrote the first and the third versions, while TSN wrote the second version. Both authors read and approved the final manuscript. Author details 1 Department of Mathematics, Wichita State University, Wichita, KS 67260‑0033, USA. 2 Department of Mathematics, California State University San Marcos, San Marcos, CA 92096‑0001, USA. Acknowledgements We thank the referee for his comments and suggestions that greatly improved the paper. We thank B. Fridman, N. Lev- enberg, J. Ribón and A. Sadullaev for helpful discussions. We are very grateful to E. Poletsky for patiently answering our many questions. Received: 10 March 2015 Accepted: 16 March 2015 References 1. Abhyankar, S.S., Moh, T.T.: A reduction theorem for divergent power series. J. Reine Angew. Math. 241, 27–33 (1970) 2. Bedford, E., Taylor, B.A.: Plurisubharmonic functions with logarithmic singularities. Ann. Inst. Fourier (Grenoble) 38(4), 133–171 (1988) 3. Deny, J.: Sur les infinis d’un potentiel. C. R. Acad. Sci. Paris Sér. 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