We have a question: Let
be homogeneous polynomials of same degree
in
. Prove that if the Jacobian
on
, then
have a common zero
?
This is an answer:
Let
be a variety defined by
. If
, then
. Therefore,
have a common zero
. So, we may assume that
. Denote by
the set of all smooth points in
. Let us consider an arbitrary smooth point
. Without loss of generality, we may assume that there exist a neighborhood
of
in
and a homeomorphism
such that
is a chart on
and
on
. It means that
for every
and for every
.
Let
and
for every
. Using Euler’s formula and note that

,
,
and
for
,
, it follows from
on 
that

Since
on
,

Similarly, we also have

Therefore, there exists a constant
such that
for every
, that is,
for all
. Since
is an abitrary smooth point in
,
for all
. Hence,
on
since
is dense in
. Consequently, there is a common zero
of
. This completes the proof.
PS. We would like to thank Prof. SIU Yum Tong for showing the way to prove this problem!
Open Problem: are
linearly independent? (it is true for case n=2)