= Zero power sums force a finite complex multiset to vanish
{title2=$\sum_{j=1}^n\lambda_j^k=0\ (1\le k\le n)\ \Longrightarrow\ \lambda_j=0$}
Collect the distinct nonzero values among the $\lambda_j$ as $\mu_1,\ldots,\mu_r$ with positive multiplicities $m_j$. The first $r$ power sums imply $\sum_jm_j\mu_j^k=0$ for $1\le k\le r$. The coefficient matrix is an invertible weighted <Vandermonde matrix>, so all $m_j$ vanish, a contradiction unless $r=0$. Its invertibility also follows because a <polynomial> $\sum_{k=1}^r a_kz^k$ vanishing at zero and all the $\mu_j$ has $r+1$ distinct roots and degree at most $r$.
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