How to prove this symmetric polynomial equations?

Question

I got a problem from a friend, which is to prove that $\Sigma _{i=1}^{n}% \frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=0$ for m < n-1.

I tried to multiply the left of equation with $\Pi _{1\leq i<j\leq n}(x_{i}-x_{j})$, and get $\Pi _{1\leq i<j\leq n}(x_{i}-x_{j})\Sigma _{i=1}^{n}\frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=\Sigma _{i=1}^{n}\{(-1)^{n-i}x_{i}^{m}\Pi _{p\neq i,1\leq p<j\leq n}(x_{p}-x_{j})\}$

However starting from here I can not conclude that this always equal to zero. I think what I am trying to do is to relate it to some sort of symmetric polynomial properties. But I couldn't figure out how.

Any hints on this? Thanks a lot.

Answer 1

You're basically talking about the kernel of the Van Der Monde matrix; I just read abut it at https://mathoverflow.net/questions/49255/how-to-determine-the-kernel-of-a-vandermonde-matrix, which may be helpful.