On Convergence Sets of Formal Power Series

The (projective) convergence set of a divergent formal power series $f(x_{1},...,x_{n})$ is defined to be the image in $\PP^{n-1}$ of the set of all $x\in \mathbb{C}^{n}$ such that $f(x_{1}t,...,x_{n}t)$, as a series in $t$, converges absolutely near $t=0$. We prove that every countable union of closed complete pluripolar sets in $\PP^{n-1}$ 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 results of A. Sathaye, P. Lelong, N. Levenberg and R.E. Molzon, and of J. Rib\'{o}n are thus generalized.


Introduction
A formal power series f (x 1 , . . . , x n ) with coefficients in C is said to be convergent if it is absolutely convergent in some neighborhood of the origin in C n . A classical result of Hartogs (see [Hartogs 1906]) states that a series f converges if and only if it converges along all directions ξ ∈ P n−1 , i.e., f ξ (t) := f ( ξ 1 t, . . . , ξ n t) converges, as a series in t, for all ξ ∈ P n−1 . This can be interpreted as a formal analog of Hartogs' theorem on separate analyticity. Since a divergent power series still may converge in certain directions, it is natural and desirable to consider the set of all such directions. Following Abyankar-Moh [Abhyankar and Moh 1970], we define the convergence set of a divergent power series f to be the set of all directions ξ ∈ P n−1 such that f ξ (t) is convergent. For the case n = 2, P. Lelong [Lelong 1951] proved that the convergence set of a divergent series f (x 1 , x 2 ) is an F σ polar set (i.e. F σ set 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 A. Sathaye (see [Sathaye 1976]) 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 . In this paper we prove that a countable union of closed complete pluripolar sets in P n−1 is the convergence set of some divergent series. This generalizes the results of P. Lelong, Levenberg and Molzon, and Sathaye. We also study convergence sets of power series of the type f (s, t) = j P j (s)t j where the coefficients P j (s) are polynomials with deg (P j ) ≤ j, as in [Ribón 2004] and [Pérez-Marco 2000].
The limit d(E) := lim k d k (E) exists ( [Zaharjuta 1975]), and is known as the transfinite diameter of E.
Let P k (C n ) be the set of polynomials on C n of degrees ≤ k. For a compact set E ⊂ C n and p ∈ P k (C n ), set The quantity c(E) is called the capacity of E.
Lemma 2.2. (Bernstein's inequality) Let E be a compact set in C n with c(E) > 0. Then there is a positive constant C E such that for every polynomial p(z) = |α|≤d a α z α , we have |a α | ≤ C d E |p| E .

Some Classes of Pluripolar Sets
Let E be a Borel subset of C n . (Though we do not mention the word "Borel" each time, all subsets of C n considered in this paper are assumed to be Borel.) The set E is said to be pluripolar (polar when n = 1) if for each point x ∈ E there is a nonconstant plurisubharmonic function u defined in a neighborhood U of x in C n such that u = −∞ on E ∩ U. The set E is said to be globally pluripolar if there is a nonconstant plurisubharmonic function u defined on C n such that E ⊂ {y : u(y) = −∞}. Josefson's theorem (answering a question of P. Lelong) states that E is pluripolar if and only if E is globally pluripolar.
The set E is said to be complete pluripolar if there is a non-constant plurisubharmonic function u defined on C n such that E = {y : u(y) = −∞}. So the set {(0, x 2 ) ∈ C 2 : |x 2 | < 1} and its closure are pluripolar, but not complete pluripolar. A countable union of pluripolar sets is pluripolar. So the set of rationals in the interval [0, 1] is polar. It is not complete polar, because each complete pluripolar set is G δ . In C each G δ polar set is complete polar, which is Deny's theorem (see [Deny 1947]).
Following Siciak [Siciak 1982, P. 2], we consider families L, G, H of plurisubharmonic functions: It follows from the one-to-one correspondence (see [Siciak 1982, Prop. 2.7]) (1) between functions of the class H of n + 1 variables and the functions of the class L of n variables that each H-complete pluripolar set in C × C n induces a unique L-complete pluripolar set in C n , and that each L-complete pluripolar set in C n is induced by a (not necessarily unique) H-complete pluripolar set in C × C n . Let |x| = (|x 1 | 2 +· · ·+|x n | 2 ) 1/2 . Recall that P k (C n ) is the set of polynomials on C n of degrees ≤ k. Let H k (C n ) be the set of homogeneous polynomials on C n of degree k. Let Definition 3.1. Let F ⊂ C n , F = ∅, x ∈ C n , and 0 ≤ r ≤ 1. Define For the empty set, we define τ H (x, ∅, r) = τ L (x, ∅, r) = 0, and T H (x, ∅) = T L (x, ∅) = 1.
where the supremum is taken over all (h, k For sufficiently small λ, since as λ 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 λ, . Letting λ → 0, and then η → 0, yields that τ (x, E ∩ K, r) = 0. Since this holds for every r < g(x)(1+|x| 2 ) −1/2 , it follows that T L (x, E∩K) ≥ g(x)(1 + |x| 2 ) −1/2 . Definition 3.5. A pluripolar set E in C n is said to be J-complete if for each x ∈ C n \ E, and each compact set K, T L (x, E ∩ K) > 0.
Note that the empty set is J-complete. Also, it is clear that a Jcomplete pluripolar set has to be closed.
Proposition 3.6. Every closed L-complete pluripolar set in C n is Jcomplete.
Proof. This is a consequence of Lemma 3.4.
Let F 1 , . . . , F m be J-complete pluripolar sets in C n and let F = ∪ m j=1 F j . Let K be a compact set in C n and let x ∈ C n \ F . Choose a number r such that 0 < r < min j T L (x, F j ∩K). Then τ L (x, F j ∩K, r) = 0 for j = 1, . . . , m. Let ε > 0. Then there are (h j , k j ) ∈ Q(C n ), j = 1, . . . , m, such that Raising each h j to a suitable power, we may assume that k 1 = · · · = k m = k. Let h = Πh j . Then (h, mk) ∈ Q(C n ), and Thus τ L (x, F ∩ K, r) ≤ ε 1/m for each ε > 0. It follows that τ L (x, F ∩ K, r) = 0 and T L (x, F ∩ K) ≥ r > 0. Therefore, F is J-complete.
The following theorem is due to A. Saddulaev, Since his book that includes the theorem has not been published, we include his proof here. We are grateful to him for sending us the statement and proof of the theorem, and to B. Fridman for translating an explanation message of A. Saddulaev from Russian to English.
Theorem 3.8. Every complete pluripolar set in C n is L-complete.
Proof. Suppose that E is a complete pluripolar set in C n . Let u be a plurisubharmonic function such that E = {x : u(x) = −∞}. Choose an increasing sequence {M j } of positive numbers such that lim M j = ∞ and M j ≥ sup |z|≤exp 2 j u(z). For each j, define a function v j by Since for each ζ on the boundary of the ball B(0, exp 2 j ), lim sup |x|<exp 2 j ,x→ζ the function v j is plurisubharmonic on C n by the gluing theorem. On each open set with compact closure, all but a finite number of v j are non-positive. It follows that the sum v(x) := ∞ j=1 v j (x) is plurisubharmonic (or identically −∞), since the sequence of the partial sums of the series is eventually non-increasing. It is clear that Suppose that y ∈ E. Then u(y) = −∞, and v j (y) = 2 −j log |y| − 1 for each j. Thus v(y) = −∞. Now suppose that y ∈ C n \E so that u(y) > −∞.
Theorem 3.9. Every closed complete pluripolarset in C n is J-complete.
Proof. This is a consequence of Proposition 3.6 and Theorem 3.8.
Let E be a non-empty compact pluripolar set in C n . Define the extremal function Φ E : The G-hull of the empty set is defined to be the empty set. SinceÊ G = ∪ k {x : Φ E (x) ≤ k}, it follows thatÊ G is an F σ pluripolar set.
A pluripolar set is said to be G-complete if it is the G-hull of a compact pluripolar set. A compact complete pluripolar set K in C n is G-complete, sinceK G = K (see [Levenberg and Molzon 1988]).

Convergence Sets in Affine Spaces
Consider a series f ∈ C[s 1 , . . . , s n ] [[t]] of the form f (s, t) = ∞ j=0 P j (s)t j , where P j (s) = P j (s 1 , . . . , s n ) are polynomials of n variables. Define Conv(f ) = {s ∈ C n : f (s, t) converges as a power series in t}.

Let A, B be nonnegative integers with
Then g is in Class (1, 0) and Conv(g) = Conv(f ). Therefore, the convergence sets for Class (A, B) are exactly the convergence sets for Class (1, 0).
Suppose that f (s, t) = ∞ j=0 P j (s)t j is in Class (1, 0) and Conv(f ) = C n . Then, by Hartogs' classical theorem, f (s, t) converges as a power series in n + 1 indeterminants s and t, i.e., f (s, t) converges absolutely for (s, t) in some neighborhood of the origin in C n × C. In this case, we say f is a convergent series. Conversely, if Conv(f ) = C n , then f (s, t) diverges as a power series in s and t, i.e., f (s, t) converges absolutely in no neighborhood of the origin in C n+1 . In this case, we say f is a divergent series.
Definition 4.1. A subset E of C n is said to be a convergence set in C n if E = Conv(f ) for some divergent series f of Class (1, 0). Theorem 4.2. Let E be a convergence set in C n . Then E is a countable union of G-complete pluripolar sets. Hence E is an F σ pluripolar set.
Suppose, if possible, that for some positive integer m, c(E m ) > 0. Then, by Bernstein's inequality (Lemma 2.2), the coefficients b jα of P j (s) = b jα s α satisfy |b jα | ≤ (C Em m) j , where C Em is a constant depending only on E m . It follows that the series f (s, t) is convergent, contradicting the hypothesis. Therefore each E m is pluripolar, and E is an F σ pluripolar set.
Fix a non-empty E m and a point s ∈Ê G m . Then γ := Φ Em (s) < ∞. Then |P j (s)| 1/j ≤ γm for all j, and hence s ∈ Conv(f ). ThusÊ G m ⊂ E for all m. Therefore, E = ∪Ê G m , and E is a countable union of Gcomplete pluripolar sets.
Theorem 4.3. Every G-complete pluripolar set in C n is a convergence set.
Proof. The theorem is proved by following the approach in [Levenberg and Molzon 1988, Theorem 5.6]. Let E be a non-empty Gcomplete pluripolar set in C n . Then E =K G , where K is a non-empty compact pluripolar set. Let F K be the collection of members (p, k) ∈ Q(C n ) such that k ≥ 1, p has rational coefficients, and |(p, k)| K ≤ 1. Let {(p j , k j )} be an enumeration of F K . Choose a sequence {r j } of positive integers so that the sequence {r j k j } is strictly increasing. Let f (s, t) = ∞ j=1 p j (s) r j t r j k j . Then f is of Class (1, 0). Suppose s ∈ E. Then α := Φ K (s) < ∞. It follows that |p j (s) r j | ≤ α r j k j for all j, and hence s ∈ Conv(f ). Therefore, E ⊂ Conv(f ).
We now consider a point s ∈ E. Then Φ K (s) = ∞. For each positive integer m there is a (p, k) ∈ Q(C n ) such that |(p, k)| K ≤ 1 and |(p(s), k)| > m, so there is a j m such that |(p jm , k jm )| K ≤ 1 and |(p jm (s), k jm )| > m. It follows that the sequence {|(p j (s) r j , r j k j )|} is unbounded, and s ∈ Conv(f ). Therefore, E = Conv(f ).
Theorem 4.4. Let E be a countable union of J-complete pluripolar sets in C n . Then E is a convergence set.
Proof. The set E can be expressed as E = ∪E m , where {E m } is an ascending sequence of J-complete pluripolar sets. For each positive integer m, we shall construct a sequence { (h mk where B m is the closed ball in C n of center 0 and radius m. Fix m and suppose that y ∈ C n \ E m . Then T L (x, E m ∩ B m ) > 0. Thus there is a positive number r < 1 such that Choose a positive rational number β = a/b < 1, where a, b are positive integers, such that (r/m) β > 1/2. There is a member (p, v) of Q(C n ) such that , and q (y) = bv. Then (h (y) , q (y) ) ∈ Q(C n ), and |(h (y) (x), q (y) )| = |(p(x), v)| β m 1−β . We have, for all x ∈ C n , , q (y) )| > m/2}. Then U y is an open neighborhood of y. Since the set C n \ E m is open, the open cover {U y : y ∈ C n \E m } of C n \E m contains a countable subcover {U y k : k = 1, 2, . . . }. Write (h mk , q mk ) = (h (y k ) , q (y k ) ). Then the sequence {(h mk , q mk )} ∞ k=1 satisfies (i), (ii) and (iii).
Theorem 4.5. Every countable union of closed complete pluripolar sets in C n is a convergence set.
Corollary 4.6. Every countable union of proper analytic varieties in C n is a convergence set.
Corollary 4.7. Every countable set in C n is a convergence set.
Corollary 4.8. A subset of C is a convergence set if and only if it is an F σ polar set.
Proof. This is because each closed polar set in C is a complete polar set.

Convergence Sets in Projective Spaces
For a formal power series f (x 1 , . . . , ]. Since for λ ∈ C, λ = 0, the series f x and f λx converge or diverge together, the convergence set of f ( i.e. the set of x for which f x converges) can be identified with a subset of the projective space P n−1 .
For a non-zero member x in C n , [x] denotes its image in P n−1 . For a subset E of P n−1 , putẼ = {x ∈ C n : [x] ∈ E}.
The (projective) convergence set of f is defined to be Definition 5.1. A subset E of P n−1 is said to be a convergence set in P n−1 if E = Conv p (f ) for some divergent series f (x 1 , . . . , x n ).
Let E be a non-empty closed set in P n−1 . Define Ψ E : The G-hull of the empty set is defined to be the empty set. If E is non-pluripolar, then E G = P n−1 . If E is pluripolar, thenÊ G is an F σ pluripolar set.
Recall that there are no non-constant plurisubharmonic functions on P n−1 .
The proofs of the following two theorems are very similar to those of Theorems 4.2 and 4.3, and hence are omitted.
Theorem 5.3. Let E be a convergence set in P n−1 . Then E is a countable union of G-complete pluripolar sets. Hence E is an F σ pluripolar set.
Theorem 5.4. Every G-complete pluripolar set in P n−1 is a convergence set.
The set Π of all hyperplanes in P n−1 is naturally isomorphic to P n−1 .
Each Ω ∈ Π is isomorphic to P n−2 , and its complement in P n−1 is isomorphic to C n−1 . For any two hyperplanes in P n−1 , there is a unitary transformation that maps one to the other.
Fix a positive number M. Let . . , 0]}, and for k = 2, . . . , n, Then {K m } is an ascending sequence of closed sets with P n−1 = ∪ ∞ m=1 K m . Definition 5.5. A subset E of P n−1 is said to be non-occupying if there exists Ω ∈ Π such that E ∩ Ω = ∅.
Lemma 5.6. If K is a closed non-occupying subset of P n−1 and if u ∈ P n−1 , then K ∪ {u} is non-occupying.
Proof. Let R = {V ∈ Π : V ∩ K = ∅} and S = {V ∈ Π : u ∈ V }. Then R is a non-empty open set in Π and S is a hyperplane in Π. Thus R \ S is non-empty.
Lemma 5.7. For each M > 0, the set K M is non-occupying.
Proof. Let e 1 , . . . , e n be the standard basis of C n , and let ε be a sufficiently small positive number. Let v j = e j + εe j+1 for j = 1, . . . , n − 1. Put V j = span(v 1 , . . . , v j ) for j = 1, . . . , n − 1, and V = V n−1 . Also, let W j = span(e 1 , . . . , e j ). Note that For j ≥ 2 and for sufficiently small ε, since W j−1 ∩ S j = ∅, and since V j−1 is close to W j−1 , we see that Definition 5.8. A pluripolar set E in P n−1 is said to be J-complete if for each hyperplane V , E \ V is J-complete in P n−1 \ V . The set E is said to be globally J-complete if for each [x] ∈ P n−1 \ E, T H (x,Ẽ) > 0.
It is clear that each J-complete pluripolar set is closed, and that each globally J-complete pluripolar set is J-complete.
The proof of the following proposition is very similar to that of Proposition 3.7, and hence is omitted.
Proposition 5.9. An intersection of (globally) J-complete pluripolar sets in P n−1 is (globally) J-complete. A finite union of (globally) Jcomplete pluripolar sets in P n−1 is (globally) J-complete.
Proposition 5.10. Let E ⊂ P n−1 be the zero locus of a continuous function h ∈ H(C n ). Then E is a globally J-complete pluripolar set in P n−1 .
Proof. This is a consequence of Lemma 3.2.
Theorem 5.11. Every closed complete pluripolar set in P n−1 is Jcomplete.
Proof. This is a consequence of Theorem 3.9.
Proposition 5.12. A proper algebraic variety in P n−1 is a global Jcomplete pluripolar set.
Let h = k j=1 |h j | 1/q j . Then h ∈ H(C n ), h is continuous, and E = {h = 0}. By Proposition 5.12, E is globally J-complete.
For [x] ∈ P n−1 and S ⊂ P n−1 , we define T H ([x], S) to be T H (x,S). If W is a hyperplane in P n−1 , and if z and S lie in P n−1 \ W ∼ = C n−1 , we observe that T H (z, S) = 0 if and only if T L (z, S) = 0.
Lemma 5.13. Let E be a J-complete pluripolar set in P n−1 , let K be a non-occupying closed set in P n−1 , let [y] ∈ P n−1 \ E, and let m be a real number ≥ 1. Then there exists an (h, q) ∈ Γ(C n ) such that (h, q) ≤ m, (h, q) E∩K ≤ 1, (h(y), q) > m/2.
Theorem 5.15. Every countable union of closed complete pluripolar sets in P n−1 is a convergence set.
Proof. This is a consequence of Theorems 5.14 and 5.11.
Corollary 5.16. Every countable union of proper algebraic varieties in P n−1 is a convergence set.
Corollary 5.17. Every countable set in P n−1 is a convergence set.
Corollary 5.18. A subset of P 1 is a convergence set if and only if it is an F σ polar set.