tailieunhanh - Báo cáo " Local polynomial convexity of union of two graphs with CR isolated singularities"

We give sufficient conditions so that the union of two graphs with CR isolated singularities in C2 is locally polynomially convex at a singularly point. Using this result and some ideas in previous work, we obtain a new result about local approximation continuous function. 1. Introduction ˆ We recall that for a given compact K in Cn , by K we denote the polynomial convex hull of K ., ˆ K = {z ∈ Cn : |p(z)| ≤ p K for every polynomial p in Cn }. ˆ We say that K is polynomially convex if K = K | VNU Journal of Science Mathematics - Physics 24 2008 6-10 Local polynomial convexity of union of two graphs with CR isolated singularities Kieu Phuong Chi Department of Mathematics Vinh University Nghe An Vietnam Received 26 October 2007 received in revised form 4 December 2007 Abstract. We give sufficient conditions so that the union of two graphs with CR isolated singularities in C2 is locally polynomially convex at a singularly point. Using this result and some ideas in previous work we obtain a new result about local approximation continuous function. 1. Introduction We recall that for a given compact K in C by K we denote the polynomial convex hull of K . K z E C p z p K for every polynomial p in C . We say that K is polynomially convex if K K .A compact K is called locally polynomially convex at a E K if there exists the closed ball B a centered at a such that B a n K is polynomially convex. A smooth real manifold S c C is said to be totally real at a E S if the tangent plane TS a of S at a contains no complex line. A point a E S is not totally real that will be called a CR singularity. By the result of Wermer if K is contained in totally real smooth submanifolds of C2 then K is locally polynomially convex at all point a E K see 1 chapter 17 . Note that union of two polynomially convex sets which can be not polynomially convex set. Let D be a small closed disk in the complex plane centered at the origin and M1 z z z E D M2 z z p z z E D where p is a C1 function in neighborhood of 0 p z o z . Then M1 M2 are totally real locally contained in a totally real manifold so that M1 M2 are locally polynomially convex at 0. The local polynomially convex hull of M1 u M2 is essentially studied by Nguyen Quang Dieu see 2 3 . Let X1 z z z E D X2 z z p z z E D where n 1 is interger and p is a C1 function in neighborhood of 0 p z o z . If n 1 then X1 and X2 is not totally real at 0 so we can not deduce that X1 and X2 are locally polynomially at 0 by the Wermer s work. .

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