It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is \begin{equation} 1 + n + C^2_n + \cdots + C^k_n \end{equation} Is there similar result for $n$ degree $d$ hypersurfaces (i.e. defined by degree $d$ polynomial, not necessarily homogeneous)? Or even more, is there any result for $n$ surfaces with multiple degrees?
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$\begingroup$ It is some how similar to the first part of the Hilbert 16th problem $\endgroup$– Ali TaghaviCommented Sep 7, 2023 at 7:19
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$\begingroup$ What is $C_n^k$? is it the binomial coefficient $\binom{n}{k}$? $\endgroup$– YCorCommented Sep 7, 2023 at 9:22
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$\begingroup$ @YCor, yes,it is $\endgroup$– Hao YuCommented Sep 8, 2023 at 8:32
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$\begingroup$ Do you know for a single hypersurface ($n=1$)? For instance $xy=1$ divides the Euclidean plane into 3 components (here $d=2$, $k=2$). $\endgroup$– YCorCommented Sep 8, 2023 at 8:55
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1$\begingroup$ There is Milnor-Thom theorem bounding the total number of faces of the arrangement. This must be, for example, in Matousek's book Lectures on Discrete Geometry. I do not have it with me now. Thus for a lack of better reference, here is a PDF containing the statement (Thm. 6.2.1). kam.mff.cuni.cz/~matousek/kvg1-tb.pdf Of course, counting all faces is probably an overkill. But at least it provides some bound. $\endgroup$– Martin TancerCommented Sep 10, 2023 at 10:15
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