### Abstract

We show that the auto Igusa-zeta function ζC,p(t) of a plane curve C over an algebraically closed field k is rational away from points p ? C of wild ramification-i.e., it is of the form f(t)/g(t) where f(t) ε Gr(Var _{k} )[L ^{-1} , t], where Gr(Var _{k} ) is the Grothendieck ring of varieties, and g(t) = Π ^{n} _{i} =1(1 - L ^{ai} t ^{bi} ) with a _{i} ε ℤ and b _{i} ε ℕ \ {0}, where L := [A1/k] is the Leftshetz motive. As a consequence, we give a new characterization for a curve C on a smooth surface S to be smooth at a point p on C when the ground field is algebraically closed and of characteristic zero.

Original language | English (US) |
---|---|

Pages (from-to) | 1825-1838 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 5 |

DOIs | |

State | Published - May 2019 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

## Fingerprint Dive into the research topics of 'The auto igusa-zeta function of a plane curve singularity is rational'. Together they form a unique fingerprint.

## Cite this

*Proceedings of the American Mathematical Society*,

*147*(5), 1825-1838. https://doi.org/10.1090/proc/13183