On irreducibility of an analytic set

Kenkiti KASAHARA
1963 Journal of the Mathematical Society of Japan
§ 1. Let D be a domain in the n-dimensional complex Euclidean space Cn, and M be a k-dimensinal analytic set" in D (1 <_ k < n-1). It is wellknown that the set of all irreducible points of M is not always an open subset of M. For example, the analytic set {zi-z2z3 = 0} in C3 is irreducible at the origin, but there exist reducible points of the analytic set converging to the origin (Osgood ). We shall say that a point p is a singular irreducible point of M, if M is irreducible at p and there
more » ... ble at p and there exist reducible points of M converging to p. Let S be the set of all singular irreducible points of M. Recently S. Hitotumatu  has shown that S must be empty if M is an analytic set of 1-dimension in C2. In this note, we show the following: THEOREM. The closure S of S in D is an analytic set in D. For each point p E S, a relation dimpS < dimpM-2 holds. REMARK. For the set S itself, Theorem is not true. For example, the analytic set {z4(4-z2z3) = 0} in C4 has the set {z1= z2 = z3 =0, z4 * 0} as S. For another example, the analytic set {z4-2z3z4+4(1-4z2) = 0} in C4 is irreducible in C4. Outside the set {z2 = 0} U {z3 = 0} U { 1-zlz2 = 0}, the analytic set is decomposed into the following four sets : {z4 = z3~1-z1V'z2 } and {z4 = --z3v"1-z1/z2 } . We have easily S={z1=z2=0,z3=z4}U{z1=z2=0,z3=-z4}-{(0,0,0,0)}. But we can generally show that the set S itself has an analytic property" that is, S is locally the finite union of locally analytic sets. (cf. § 4.) First applying the Remmert-Stein's ' Einbettungssatz' () and the method of Osgood [2, Chap. II, § 15], we shall define the number of components of M at a point p E M. (cf. § 2). In § 3, we shall derive a property of roots of a polynomial. In § 4, we shall consider Theorem for the case that M is.