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Let $V$ be a vector space with a basis $f_1, \ldots, f_n$. Here $f_i$ are complex functions on $\mathbb{C}$. Defined the sequence of exponents of $V$ at $\lambda \in \mathbb{C}$ as follows. There is a basis $g_1, \ldots, g_n$ of $V$ and a sequence of integers $c_1<\cdots such that g_i(\lambda)=0, g'_i(\lambda)=0, \ldots, g_i^{(c_i-1)}(\lambda)=0, g_i^{(c_i)}(\lambda) \neq 0 (i=0, \ldots, n). It is said that $c_1, \ldots, c_n$ is unique and does not depend on the choice of a basis of $V$. How to show that $c_1, \ldots, c_n$ is unique? Thank you very much.

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    What have you tried? If you have a different basis $(h_i)_i$ and integers d_1<\cdots that do the same thing for $h$ as the $c$'s for the $g$'s, then you must show that $d_1=c_1$, $d_2=c_2$, etc. Perhaps by induction on $n$? (Note without extra assumptions -- such that all functions are holomorphic in a neighborhood of $\lambda$ -- it would be possible that there is no matching $c_i$ sequence at all, except that the exercise specifies as a premise that there is at least one).2011-09-28

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