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There are some theorems about various zeta functions which states the rationality of those. For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ for $f\in\mathbb{Z}[x]$ -i.e. $P(T)=\sum \#\{x \ mod \ p^n ; f(x)=0 \}T^n$, it is rational. However, function $P$ is supposed to be more complicated when $f$ is more singular. It is an experience and it is what the proof of rationality somehow says.

My question is what are known methods to measure a complexity of rational functions or a singularity of varieties?

One can think about using a Galois something, lengths of blowups, norms... I'm really intrested what has been developed in this direction so far.

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    $\begingroup$ Motivic zeta functions are another example. You look at the varieties X(k[t]/t^n), aka truncated arc spaces, and see how these vary with n. I believe this goes back to Nash. See numerous papers of Denef and Loeser for (many!) results, relations to vanishing cycles etc $\endgroup$
    – user108998
    Commented Dec 1, 2019 at 19:53
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    $\begingroup$ I should mention that there is a pretty massive theory of vanishing cycles, which can be a bit formidable. See papers of Sabbah for an intro. A way I really like to measure singularities is in terms of the twisted de rham complex, ie you take forms and the differential d+df (for a function f) $\endgroup$
    – user108998
    Commented Dec 1, 2019 at 19:57

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To my knowledge, multiplier ideals are the finest way to "measure" singularities. If all the singularities are in the base field, then multiplier ideals are also intimately related with the information on the poles of the Igusa zeta function. Here one short reference for an introduction to the topic http://library.msri.org/books/Book51/files/03blickle.pdf .

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