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
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)