ON CONVERGENCE SETS OF FORMAL POWER SERIES

Аннотация

This work is devoted to the study of sets of convergence of divergent formal power series of many variables. In this note we study the region of convergence of formal power series with holomorphic coefficients. 1 INTRODUCTION A set E is called complete pluripolar if there exists a non-constant plurisubharmonic n function This work is devoted to the study of sets of u such that convergence of divergent formal power series of many variables. In this note we study the region of E ={z:u(z)=−}. convergence of formal power series with holomorphic coefficients. 2 Materials and Methods A formal power series f (z ,...z ) is called 1 n Consider a formal power series of the form convergent if this series converges in some neighborhood of the origin in n. The classical f (z,t)=  P (z) tj, (1) result of Hartogs states (see [1]) that a series f j j=0 converges if and only if it converges along all where P (z)= P (z ,...,z )are polynomials of j j 1 n sections P n−1, i.e. f (t)= f ( 1 t,... n t) n variables. Define converges as a series in a variable t for all Conv(f)={z n: f(z,t) converges as a power series in t}. P n−1. This result can be interpreted as a formal analogue of Hartogs' theorem on separate Let A and B be nonnegative integers such that A> 0. analyticity. Since a divergent power series can Series (1) is said to belong to the class Class(A, B) converge in some sections, it is naturally of interest to if degP  A j+B. j study and describe such sections. For example, a series f (z , z ) = k!z k1 z k2 Clearly, Class (1, 0) is a subclass of Class(A, B). 1 2 1 2 Suppose that that for some k=0 E =Conv(f ) is divergent, but on complex lines z 1 = 0and f Class(A,B). Let f(z,t)=  P(z)tj j z = 0 it converges. j=0 2 and g(z,t)=tN f (z,tN), where N=A+B. Then Let E a Borel set in n. A set E is called gClass(1,0) and Conv(g)=Conv(f). pluripolar if for each point zE there is a non- Therefore, the convergence set Class(A, B) coincides constant plurisubharmonic function u(z) defined in with the convergence set Class(1,0). some neighborhood U  n of the point z such Suppose that a series that u =− on EU . A set E is called  is from the class (1, 0) globally pluripolar if there exists a non-constant f (z,t)=P (z) tj, j plurisubharmonic n function u(z) such that j=0 and Conv(f)= n. Then, according to Hartogs’ E {z:u(z)=−} . Josephson's theorem fundamental theorem, the series f (z,t) converges answers the question posed by P. Lelon and states as a power series in the n+1 variables z and t, i.e. that it is pluripolar if and only if it is globally pluripolar (see, for example, [2,3]). f (z,t)converges absolutely in the variables (z,t) in some neighborhood of the origin in n . In 1 this case, we say that f it is a convergent series. On By Bernstein’s inequality (3), the coefficients b of the polynomial satisfy the contrary, if Conv(f) n , then f (z,t) j P (z)=b z j j diverges as a power series in the variables z and t,  the inequality i.e. does not converge absolutely in any neighborhood of the origin in n+1 , then we say b (c )j, that f it is a divergent series. j Em c where is a constant depending only on E . It E m m Definition 2.1. A set E nis called a set of follows that the series f (z,t)converges. We have convergence in n if E=Conv(f) for some obtained a contradiction to the fact that the series is divergent series of class (1, 0). divergent. Therefore, for each m the set E is m pluripolar and Eis a F pluripolar set.► In the work [4] Daowei Ma and Tejinder S.  Neelon, the following statement was proven. 3 Main results Theorem 2.1. Let E the set of convergence in n. Then E is a countable union of G – complete In this paper, we consider a more general pluripolar sets. Therefore, Eit is a F pluripolar situation, namely, we consider a divergent formal set. power series of the form Let us present a diagram of the proof of this  f (z,t)=c (z) tj (4) theorem and show what Ea F pluripolar set is. j  j=0 Consider the divergent series n with holomorphic coefficients of the variables and the set of convergence of series of the form (4) is f (z,t)=P (z) tj studied. j j=0 Give let a formal power series (4), where the class (1, 0) such that E =Conv(f ) . Let's put coefficients c (z)=c (z ,...,z ) are j j 1 n holomorphic functions in some complete circular  1  E =z n: z m, P(z) j m for j=1,2,...  domain D n with respect to the origin. The m j   main result of the work is the following theorem. for m=1,2,.... (2) Theorem 3.1. If E is the convergence set of Then  . Let us prove that E they are some divergent series (5), then E the set is a E = E m m pluripolar set. m=1 pluripolar sets. It should be noted that the method of proving Theorem 2.1 does not work when proving Theorem Let us assume the opposite, let for some positive 3.1, since the proof of Theorem 2.1 essentially uses integerm,c(E )0. Now we will use the m the fact that the coefficients of the series are following Bernstein lemma (see