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Better wording of definition
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aisejohan committed May 16, 2024
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Expand Up @@ -903,10 +903,11 @@ \section{Algebraic multiplicities}

\begin{definition}
\label{definition-multiplicity}
In the situation above, if $d \geq \dim(\text{Supp}(M))$, then we set
$e_I(M, d)$ equal to $0$ if $d > \dim(\text{Supp}(M))$
and equal to $d!$ times the
leading coefficient of the numerical polynomial $\chi_{I, M}$ so that
In the situation above, assume $d \geq \dim(\text{Supp}(M))$.
In this case, if $d > \dim(\text{Supp}(M))$, then we set $e_I(M, d) = 0$
and if $d = \dim(\text{Supp}(M))$, then we set $e_I(M, d)$ equal to $d!$
times the leading coefficient of the numerical polynomial $\chi_{I, M}$.
Thus in both cases we have
$$
\chi_{I, M}(n) \sim e_I(M, d) \frac{n^d}{d!} + \text{lower order terms}
$$
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