4
$\begingroup$

Suppose $p$ is a homogeneous polynomial in $n$ complex variables. Let S be the hypersurface defined by $p(z)=0$. Then is the 1-form $dp/p$ always non-exact on the complement $C^n\setminus S$?

Any answer or reference is appreciated.

Ron Yang

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ Trivial objection: p(z) = 1 is a homogeneous polynomial. $\endgroup$
    – Willie Wong
    Commented Aug 24, 2010 at 15:40

1 Answer 1

Reset to default
6
$\begingroup$

Assume that $p$ is non-zero. If the form $dp/p$ were exact, then locally a primitive would be $log(p)+const$; this is easily seen not to work as soon as you can "loop around" $S$ (e.g. restrict everything to a line intersecting $S$ and see what happens there). Thus the form $dp/p$ is exact if and only if $S$ is empty, and hence if and only if $p$ is constant.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thanks for the answer. I guess the same thing can be said for all nontrivial polynomials. $\endgroup$
    – Ron
    Commented Aug 25, 2010 at 13:44
  • $\begingroup$ Yes, the fact that $p$ is homogeneous plays no role in this argument. $\endgroup$
    – M P
    Commented Aug 25, 2010 at 15:00

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .