On approximation by special analytic polyhedral pairs

Abstract

Abstract. For bounded logarithmically convex Reinhardt pairs "compact set -domain" (K, D) we solve positively the problem on simultaneous approximation of such a pair by a pair of special analytic polyhedra, generated by the same polynomial mapping f : D → C n , n = dim Ω. This problem is closely connected with the problem of approximation of the pluripotential ω(D, K; z) by pluripotentials with a finite set of isolated logarithmic singularities ([23, 24]). The latter problem has been solved recently for arbitrary pluriregular pairs "compact set -domain" (K, D) by Poletsky [12] and S. Nivoche [10,11], while the first one is still open in the general case.

On Lelong-Bremermann Lemma

“…Fix any compact set K ⊂ D. Since u is continuous, we conclude from the representation (4.1), following [18], that for each δ > 0 there is a function…”

“…The main result of this paper (Theorem 1) says that the number N in the above proposition can be taken ≤ 2n + 1, where n is the dimension of D. The proof is based on a generalization of the reduction argument given in [18], Lemma 2, combined with a perturbation argument for smooth mappings.…”

“…Another important example has been considered in [18]: using a construction similar to (2.6) in a combination with some algebraic considerations (see Lemma to n = dim D is due to the fact that the pluripotential (4.6) is a maximal plurisubharmonic function ( [15]) in the annulus D K.…”

“…The property (2.4) helps us to use an idea from [18] on how to reduce the number of functions from N to m. Namely, we construct a sequence of mappings…”

“…Proposition 1 does not say anything about the behavior of the numbers N = N (K, ε). However, information about the bound for N is welcomed in certain investigations such as attempts to approximate simultaneously a pluriregular pair by a sequence of similar analytic polyhedral pairs (see [12,13,18]). …”