What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$? 


I am interested in this question for constant $k$, say $k=3$. 


One trivial bound is $d \choose 2$, an other is $kd$.


Does there exist an example of a polynomial with more than $k+d$ zeroes in total of it and its derivatives? 


In other words, if $p$ is a polynomial and $x\_1,\ldots x\_k$ are his zeroes then I am asking how many $i,j$ there exist such that $p^{(i)}(x\_j)=0$



Number of zeros of the derivatives of a polynomial
The answer is yes. Here is an example for $k=3$. For $m>0$, take $f(z)=z^m(z^2-1)^m$. Then $d=3m$. Of course, $-1$, $0$ and $1$ are roots of $f^{(i)}$ for $0\le i\le m-1$, so we have $3m=d$ roots. However, $f^{(i)}$ is an odd function for $i=m+1,m+3,\ldots, 3m-1$, producing $m$ more roots in $0$. So in this example, the count of roots is at least $\frac{4}{3}d$, which is bigger than $d+3=d+k$ once $m>3$.


