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SUMMARY:Ashish Dwivedi (IIT Kanpur) and Nitin Saxena (IIT Kanpur)
DTSTART:20200703T140000Z
DTEND:20200703T143000Z
DTSTAMP:20260423T200314Z
UID:ANTS14/20
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ANTS14/20/">
 Computing Igusa's local zeta function of univariates in deterministic poly
 nomial-time</a>\nby Ashish Dwivedi (IIT Kanpur) and Nitin Saxena (IIT Kanp
 ur) as part of Algorithmic Number Theory Symposium (ANTS XIV)\n\n\nAbstrac
 t\nIgusa's local zeta function $Z_{f\,p}(s)$ is the generating function th
 at counts the number of integral roots\, $N_{k}(f)$\, of $f(\\mathbf x) \\
 bmod p^k$\, for all $k$. It is a famous result\, in analytic number theory
 \, that $Z_{f\,p}$ is a rational function in $\\Q(s)$. We give an elementa
 ry proof of this fact for univariate $f$. Our proof is constructive as it 
 gives a closed-form expression for the number of roots $N_{k}(f)$.\n\nOur 
 proof\, when combined with the recent root-counting algorithm of (Dwivedi\
 , Mittal\, Saxena\, CCC\, 2019)\, yields the first deterministic poly($|f|
 \, \\log p$) time algorithm to compute $Z_{f\,p}(s)$. Previously\, an algo
 rithm was known only in the case when $f$ completely splits over $\\Q_p$\;
  it required the rational roots to use the concept of generating function 
 of a tree (Zúñiga-Galindo\, J.Int.Seq.\, 2003).\n\nChairs: Marco Streng 
 and David Kohel\n
LOCATION:https://researchseminars.org/talk/ANTS14/20/
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