A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha \in \mathbb{Z}_+^n}} \gamma_{\alpha} \xi^{\alpha}$$ A root $r$ of $P(\xi)$ is of order $k$ if $$ \sum\limits_{(\alpha, \lambda) < k} |D^{\alpha}P(r)| = 0, \ \sum\limits_{(\alpha, \lambda) = k} |D^{\alpha}P(r)| \neq 0 $$ I would like to determine when $P(\xi)$ has a non-zero root of order $m$. When $n = 2$, $P(\xi)$ can be factorized into linear factors so I can use this factorization. However, this doesn't work for $n \geq 3$ as there are irreducible polynomials over the complex field. What algebraic or analytic methods can I use to approach this problem for $n \geq 3$? Can you cite any useful papers on the subject?
$\begingroup$
$\endgroup$
Add a comment
|