How to calculate "mass" or "center of mass" of a set of solutions to an algebraic equation?

Question

Let us assume we have some algebraic equation

$$f(\{x_k\}) = f(x_1,\cdots,x_N) = \sum_{i_1,\cdots,i_N \in \{0,\cdots,N\}}^{} c_{i_1\cdots i_N}\cdot{x_1}^{i_1}{x_2}^{i_2}\cdots {x_N}^{i_N} = 0$$

As a famous special case we can mention the famous equation of a circle, $N=2$:

$$x_1 = x, x_2 = y\\x^2+y^2-1=0$$

As we can verify, this circle is centered on origo, and therefore it is fair to say that origo is the "center of mass", imagining every solution having the same "density" in some sense. We can of course easily expand this to a 3-sphere or even higher dimensions.

How can we define a concept similar to that of mass in physics so that for any surface being the solutions to an equation has the same density, let us say $$\rho(\{x_k\}) = \cases{\rho \neq 0, \text{iff } f(\{x_k\}) = 0\\0\phantom{ \neq 0\,\,} \text{, iff } f(\{x_k\}) \neq 0}$$

I suppose such a definition of mass could go

$$m = \int_{\forall \{x_k\}\in R^N} \rho(x_k) d{x_k}$$

And as for center of mass

$${\bf v_c} = \int_{\forall \{x_k\}\in R^N} {\bf x} \rho(x_k) d{x_k}$$

Does this make sense, and if it does, how to approach calculating it in practice? I mean, obviously integrating over whole $\mathbb R^N$ for a particular solution space seems... really difficult.

I suppose something like the answer to this question could help us on this journey?

Given some way to step around parametrically on surface: Can we find the center?

Answer 1

Your equations for the mass and center of mass are correct. Often we have an item which is limited in extent, so you only need to integrate over the volume it occupies. You can even do it if $\rho$ varies in space. The integrals will only be analytically tractable if the function is simple. Otherwise, we can do the integration numerically, just like any other integration.