Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations: \begin{equation} P_1(\vec x)=0, \\ \vdots \\ P_m(\vec x)=0, \end{equation} (where $\vec x$ are coordinates in $\mathbf R^n$). How can the volume of $M$ be expressed in terms of the data contained in the polynomials $P_1,\dotsc,P_m$?
$\begingroup$
$\endgroup$
6
-
$\begingroup$ What do you mean by "volume"? The Lebesgue measure will be zero if $m \geq 1$. (But there is probably still a sensible way to define lower-dimensional volume...) $\endgroup$– Sam HopkinsCommented Sep 20, 2022 at 20:54
-
3$\begingroup$ @SamHopkins Volume as in $\int_M \sqrt{\det g}$ where $g$ is the induced metric. $\endgroup$– dennisCommented Sep 20, 2022 at 20:57
-
3$\begingroup$ Probably the simplest expression would be the Cauchy-Crofton theorem. $\endgroup$– Ryan BudneyCommented Sep 20, 2022 at 22:20
-
1$\begingroup$ Your question is a bit vague. Could you tell us what sort of expression you would accept as an answer when $m=1$ and $n=2$? $\endgroup$– Robert BryantCommented Sep 21, 2022 at 9:58
-
3$\begingroup$ @dennis: Good luck with that. The length of the ellipse $x^2/a^2+y^2/b^2=1$ is a non-elementary function of $a$ and $b$, expressed in terms of an elliptic integral. Getting an explicit formula for the length of, say, a bounded quartic curve in terms of the coefficients of the quartic polynomial defining it is almost certainly hopeless. $\endgroup$– Robert BryantCommented Sep 21, 2022 at 15:16
|
Show 1 more comment