The algebra generated by $z^3$ and a continuous function on a compact set in the plane
Nội dung chính của bài viết
Tóm tắt
We study uniform approximation on a compact set $K \subset \C$ by the algebra generated by $z^3$ and a continuous function $f$. The main geometric point is that, after pulling back by the polynomial map $(z,w)\mapsto (z^3,w)$, the graph of $(z^3,f(z))$ splits into three branches. If the values of $zf(z)$ stay in a sector of aperture strictly smaller than $2\pi/3$, then these three branches can be separated by the polynomial $p(z,w)=zw$. Two applications of Stout's theorem then give density of the polynomial algebra on the pulled-back set. We record an abstract cubic
criterion, a smooth version obtained from the standard one-graph theorem, and several concrete examples.
Từ khóa
polynomial convexity, totally real manifolds
Chi tiết bài viết
Tài liệu tham khảo
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