A classical result states that roots of a polynomial are continuous functions of its coefficients. This is, for exemple, a direct consequence of Rouché's theorem.

Using the implicit function theorem, one can further show that the mapping "coefficients $\mapsto$ roots" is smooth, provided one restricts to polynomials having $\textit{simple}$ roots. It is however easy to check this mapping is not $C^1$ near polynomials where two distinct roots merge to one double root (square root behavior).

I'm considering the following problem: Fix any ordering $\prec$ on $\mathbb C$ and order the zeros of a polynomial $P$ of degree $n$ as $x_1\prec\ldots \prec x_n$ (or restrict to the set of polynomials having only real roots if you prefer), and consider the set of polynomials having smaller root $y_1$ of given multiplicity $m_1$, second root of multiplicity $m_2$, etc. Thus, for some $k\geq 1$, $$ \{x_1,\ldots,x_n\}=\{\underbrace{y_1,\ldots,y_1}_{m_1},\ldots,\underbrace{y_k,\ldots,y_k}_{m_k}\},\qquad y_1\prec \ldots\prec y_k. $$ I would believe that the mapping "coefficients $\mapsto$ roots" is smooth (say at least $C^1$) once restricted to such a subclass of polynomials, for any fixed multiplicities $m_1,\ldots,m_k$, but I haven't been able to locate such a result in the literature, nor to prove it using similar strategies as in the previous statements. Any idea?

**Edit:** If you prefer matrices, I am looking for a statement like: The spectrum of Hermitian (resp. normal) matrices is smooth in the matrix coefficients provided we restrict to Hermitian (resp. normal) matrices having eigenvalues with prescribed multiplicities $m_1,\ldots,m_k$.

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