How to judge whether a polynomial have a factor of multiplicity $d$ Let $f(x)=a_nx^n+\cdots+a_0$ be a polynomial of degree $n$. To judge whether $f(x)$ has a square factor, we only need to consider the resultant $Res(f,f')$. 
My question is, if we want to judge whether $f(x)$ has a $d$-multi factor, what can we do? Of course we can consider the resultant $Res(f^{(d-2)},f^{(d-1)})$ or something similar, but I think it's too coarse. We can also use the theory of multi-resultant, but it's not explicit enough. 
If we ask the same question to a polynomial of multi variables, are they same? 
 A: Let's rehomogenize to a binary form $F(x_1,x_2)=x_2^n f
\left(\frac{x_1}{x_2}\right)$. Then (assuming $a_n\neq0 $) the question is about detecting a factorization $F=L^d G$ where $L$ is linear. Consider all multiindices $\alpha=(\alpha_1,\alpha_2)$ of length $|\alpha|=d-1$ and introduce two sets of $d$ indeterminates $s=(s_{\alpha})_{\alpha}$ and
$t=(t_{\alpha})_{\alpha}$ indexed by these multiindices. The equations you are looking for can be obtained as the coefficients of the polynomial
$$
P(s,t)={\rm Res}\left(\sum_{\alpha}s_{\alpha}\partial^{\alpha}F,\sum_{\alpha}t_{\alpha}\partial^{\alpha}F\right)
$$
which is bihomogeneous of degree $(n-d+1)$ in $s$ and in $t$.
I think the degree $2(n-d+1)$ set theoretic equations obtained in this way are about as good as one can do for $d$ small but certainly not optimal for large $d$. In that case it might be better to use equations (for the ideal) found by Weyman in a series of three articles:


*

*J. Weyman, "The equations of strata for binary forms." Journal of Algebra 122, no. 1 (1989): 244-249.

*J. Weyman, "Gordan ideals in the theory of binary forms." Journal of Algebra 161, no. 2 (1993): 370-391.

*J. Weyman, "On the Hilbert functions of multiplicity ideals." Journal of Algebra 161, no. 2 (1993): 358-369.


For $d$ high enough, ideal generators are quadratic and correspond to (easy to calculate) transvectants of $F$ with itself.
For more general coincident root loci see my article with Chipalkatti mentioned in my answer to this MO question and references therein.

Edit: the article I mentioned namely, "On Hilbert covariants" also treats the multivariate case. Theorem 6.1 therein gives set-theoretic equations for a homogeneous polynomial $F(x_1,\ldots,x_r)$ to be of the form $G^d$. The idea is to use the binary form case and intersect your hypersurface with a variable line. With classical invariant theory it is easy to see how this translates at the level of explicit equations and this device is called the Clebsch transfer principle.

Edit 2: A new article came out today with a characterization of triple roots using a "resultant of resultants". I don't yet know how this intriguing construction relates to the one I explained above.
