Multiplicity of roots of a fewnomial It is easy to show that a complex polynomial with $N$ non-zero coefficients cannot have a non-zero root of multiplicity $N$ or more. Is there some standard name / reference for this fact? Are there any (non-technical) improvements known?
Thank you!

Also asked (and answered) on MathOverflow, about a week after being originally posted here.
 A: Here is a proof.  I don't recall seeing it before, but that is not an indication of anything.
Theorem: A polynomial (not identically $0$) over a field $k$ of characteristic $0$ which has
$N$ nonzero coefficients can't have a root of multiplicity $\ge N$.
I use induction on $N$.  It's obvious for $N=1$: a polynomial with only one nonzero coefficient has no nonzero roots.  
Now suppose it's true for $N$, and consider a polynomial $P \in k[X]$ with $N+1$ nonzero coefficients.  If $P$ is divisible by $X^k$, dividing by $X^k$ doesn't change the nonzero roots or their multiplicities.  Thus we may assume wlog that one of the $N+1$ nonzero coefficients is the $X^0$ term.   Now if $r$ is a root of $P$ of multiplicity $\ge N+1$, it is a root of $P'$ of multiplicity $\ge N$, and $P'$ has $N$ nonzero coefficients.  By assumption, this is impossible.
By the way, the theorem is false if you omit the assumption that the field has characteristic $0$.  For example, over a field of characteristic $p$,
$x^p + 1 = (x+1)^p$ has two nonzero terms and a nonzero root of multiplicity $p$. 
