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Let $X/k$ be a (geometrically integral and connected) variety over $k$ either a field of characteristic $0$ or a finite field. Let $[X] = \sum_{i\in I}[Y_i] - \sum_{j\in J}[Z_j]$ be a decomposition inside the Grothendieck ring $K_0(\mathrm{var})$ with the $Y_i,Z_j$ smooth, projective and connected. It seems to me that the number $\rho(X) = |I| - |J|$ is an invariant of $X$ and does not depend on the decomposition.

Proof of independence: First, let $k$ be a characteristic $0$ field. Let $E(X)(u,v)$ be the Euler-Hodge-Deligne polynomial defined using mixed hodge structures. It factors through $K_0(\mathrm{var})$ and for a smooth,projective and connected scheme $X$, $E(X)(0,0) = 1$; therefore $\rho(X) = E(X)(0,0)$.

Second, let $k = \mathbb F_q$ be a finite field. We claim that $\rho(X)$ is the order of the pole at $t=1$ or the zeta function $$\zeta_X(t) = \exp\left(\sum_{n\geq 1}|X(\mathbb F_{q^n})|\frac{t^n}{n}\right).$$ It is well known that $\zeta_{X\sqcup Y}(t) = \zeta_X(t)\zeta_Y(t)$ and that for a smooth projective connected scheme, there is a unique pole of order $1$ (by the Weil conjectures). This completes the proof as in the first case.

Question 1. Does this invariant have a name?

Question 2. Is there an alternative proof of this independence, perhaps one that also works uniformaly over an arbitrary base?

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    $\begingroup$ Let $S$ be a smooth surface, $p\in S$, and $U:=S\smallsetminus p$. Blow up $p$, then a point above $p$, etc. to get a birational morphism $\pi :\hat{S}\rightarrow S$ such that $\pi ^{-1}(p)$ is the union of exceptional divisors $E_1,\ldots,E_n$. Then $[U]=[S]-[p]=\hat{S}-\sum[E_i]$ in the Grothendieck ring. $\endgroup$
    – abx
    Commented May 30, 2022 at 4:04
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    $\begingroup$ @abx: Not quite so, because $E_i$ intersect, and therefore you need to add the classes of intersection points to the right hand side. $\endgroup$
    – Sasha
    Commented May 30, 2022 at 5:14
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    $\begingroup$ @Sasha: Oops, right, of course. That was too early in the morning... $\endgroup$
    – abx
    Commented May 30, 2022 at 6:52
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    $\begingroup$ This holds in Bittner's blow up Grothendieck ring (isomorphic to the usual one in char. 0), as blow up relations preserve connected components. In char. p in general one could try to use etale cohomology with compact supports as this is what zeta-functions boil down to over finite fields. $\endgroup$
    – Evgeny Shinder
    Commented May 30, 2022 at 10:27

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