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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

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    $\begingroup$ Trivial objection: p(z) = 1 is a homogeneous polynomial. $\endgroup$ Aug 24 '10 at 15:40
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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.

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  • $\begingroup$ Thanks for the answer. I guess the same thing can be said for all nontrivial polynomials. $\endgroup$
    – Ron
    Aug 25 '10 at 13:44
  • $\begingroup$ Yes, the fact that $p$ is homogeneous plays no role in this argument. $\endgroup$
    – M P
    Aug 25 '10 at 15:00

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