Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that maps the area to a msphere? Bounded means the area $A<\infty$ with the boundary defined by the polynomials. To partition means to partition with kdimension manifold($k=m1$) as boundary(the intersection of the two subset) .
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1$\begingroup$ What does "partitioned" mean? A disjoint union? (This clearly can't happen.) An open cover? $\endgroup$ – Qiaochu Yuan Nov 15 '14 at 6:43

$\begingroup$ @QiaochuYuan, thank you, I will edit it to clarify the question. $\endgroup$ – XL _At_Here_There Nov 15 '14 at 6:53

1$\begingroup$ I still don't understand the question. The $n$sphere cannot even immerse into $\mathbb{R}^n$. Did you mean the $n1$sphere, or maybe the $n$disk? $\endgroup$ – Qiaochu Yuan Nov 15 '14 at 7:17

$\begingroup$ @QiaochuYuan, thank you for your patience. Let me give a example in 1dimension, $\int_a^b f(x)dx=\int_a^c f(x)dx+\int_c^b f(x)dx $, [a,b] is partitioned into [a,c],[c,b]. $\endgroup$ – XL _At_Here_There Nov 15 '14 at 7:25

1$\begingroup$ it is wellknown that any semialgebraic set can be triangulated. The question you ask seems like a particular case of this fact. $\endgroup$ – Dima Pasechnik Nov 15 '14 at 9:41
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See for instance:
S. Łojasiewicz, Triangulation of semianalytic sets, Ann. Scuola Norm. Sup. di Pisa, ser. 3, 18.4 (1964), pp. 449–474,
or start a search with the keywords "Triangulation" and "semialgebraic", for more recent results, or more suitable to your needs.

$\begingroup$ Thank you, I will read the article, and search with the keywords. $\endgroup$ – XL _At_Here_There Nov 15 '14 at 9:48

1$\begingroup$ Actually there is a lot of material online. Chapter 3 of these notes: perso.univrennes1.fr/michel.coste/polyens/SAG.pdf $\endgroup$ – Pietro Majer Nov 15 '14 at 10:25

$\begingroup$ Yes, I have just found a lot of material online. Thank you again. $\endgroup$ – XL _At_Here_There Nov 15 '14 at 10:36