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17 votes
4 answers
772 views

Partitions of $\mathbb{R}^d$ by implicit polynomial equations

Given a polynomial $p(x_1,x_2,\ldots,x_d)$ in $d$ variables, with maximum degree $k$, what is the maximum number of components of $\mathbb{R}^d$ minus $p(\ldots)=0$? In other words, into how many ...
Joseph O'Rourke's user avatar
5 votes
4 answers
1k views

The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$ Can anyone point me to a concise, modern derivation and explanation of the significance of the icosahedron equation, more modern and concise than Klein's description in ...
Joseph O'Rourke's user avatar
4 votes
0 answers
119 views

Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is $$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$ ...
M. Winter's user avatar
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