The auto igusa-zeta function of a plane curve singularity is rational

Andrew R. Stout, Lev Borisov

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1825-1838
Number of pages14
JournalProceedings of the American Mathematical Society
Issue number5
StatePublished - May 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics


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