From: instanton@mail.ru
Subject: Re: Genus of a curve - draw Newton polygon
Date: Wed, 07 Feb 2001 10:46:29 GMT
Newsgroups: sci.math
Summary: Determine genus of a nonsingular plane curve using Newton polygon

In article <95o6hv$egk$1@nnrp1.deja.com>,
  bill_pet@my-deja.com wrote:
> Is theere a formula to compute the genus of the curve p(x,y) = 0
> where p(x,y) is a polynomial of degree n ?
>
> Thanks,
> Bill
>
> Sent via Deja.com
> http://www.deja.com/
>

Draw Newton polygon
and calculate the number of integer points in it
that will be genus !

(this works may be at assumptions of nonsingulyrity
may be even without)

What is Newton polygon -
asumme p(x,y)= \sum_ij a_ij x^iy^j
and take the convex hull of the integer points (i,j)
such that a_ij >< 0
-that's it!

 and calculate number of integr points in it - this is genus


The proof: holomorphic differentials can be written
x^iy^j dx/ p'

where derivative p' with respect to y
check that if ij belong to Newton than it works

( i did not do that, sorry :)

This should be very well-known
but i do not know references it seems it is not in Grif-Har,
i took it in some russian text book which seems to be only in russian

Your's Sasha

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==============================================================================

From: "Charles Matthews" <charles@sabaki.demon.co.uk>
Subject: Re: Genus of a curve - draw Newton polygon
Date: Wed, 7 Feb 2001 11:58:27 -0000
Newsgroups: sci.math

instanton@mail.ru wrote

>> Is theere a formula to compute the genus of the curve p(x,y) = 0
>> where p(x,y) is a polynomial of degree n ?
>
[quoted passage deleted -- djr]
>
> and calculate number of integr points in it - this is genus


Interesting.  This seems to give the correct answers for the general curve
of degree n and hyperelliptic curves y^2= P(x) which have genus [(n-1)/2]
where P has degree n.  That's counting lattice points in the interior.  But
I think it fails for singular curves, eg y^2 = x^3 + x^2 which should have
genus 0.

Charles