The dlt motivic zeta function is not well defined
As its title explicitly suggests, the object of the article under review is to show that the dlt motivic zeta function introduced by \textit{C. Xu} [Mich. Math. J. 65, No. 1, 89--103 (2016; Zbl 1346.14062)] is \textit{not} well defined, hence contradicting the main claim of the aforementioned paper.\N\NMore precisely, the definition in [\textit{C. Xu}, Mich. Math. J. 65, No. 1, 89--103 (2016; Zbl 1346.14062)] \textit{does} depend on the choice of a so-called \textit{dlt modification}.\N\NThis dlt motivic zeta function, associated with a regular function \(f\) on a smooth variety X over a field of characteristic zero, is a variant of the zeta function introduced in [\textit{J. Denef} and \textit{F. Loeser}, J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)].\N\NAfter a set of preliminaries (dlt pairs, differents, stringy motives...) extracted for example from [\textit{W. Veys}, Adv. Stud. Pure Math. 43, 529--572 (2006; Zbl 1127.14004)] and [\textit{V. V. Batyrev}, J. Eur. Math. Soc. (JEMS) 1, No. 1, 5--33 (1999; Zbl 0943.14004)], the authors present a number of counterexamples to Hu's claim, starting from an explicit Newton nondegenerate polynomial\N\[\Nf = x^4 +x^2y^2 +y^6 +z^3\N\]\Nand constructing two very explicit blow-ups (with pictures of the corresponding fans for the reader's convenience) which are two different dlt modifications for the pair \(( \mathbb{A}^3 , Z ( f ) ) \) giving two different values for the motivic zeta function. Plus, another dlt modification is obtained via a flop, providing a second counter-example.\N\NThe end of the paper is devoted to an analysis of three incorrect proofs (the ones of Proposition 3.1, Proposition 3.2 and Theorem 1.2) appearing in \textit{C. Xu}'s paper [Mich. Math. J. 65, No. 1, 89--103 (2016; Zbl 1346.14062)], and to comments about how one could modify or exploit Hu's definition of the dlt motivic zeta function. Very roughly, the conclusion of the paper seems to be that a correct and exploitable definition (in particular, regarding the motivic analogue of the Monodromy Conjecture) remains to be found in general.
scientific article; zbMATH DE number 7922236