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Example 42.42.6. For every $p \geq 1$ there is a unique homogeneous polynomial $P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ]$ of degree $p$ such that, for any $n \geq p$ we have

\[ P_ p(s_1, s_2, \ldots , s_ p) = \sum x_ i^ p \]

in $\mathbf{Z}[x_1, \ldots , x_ n]$ where $s_1, \ldots , s_ p$ are the elementary symmetric polynomials in $x_1, \ldots , x_ n$, so

\[ s_ i = \sum \nolimits _{1 \leq j_1 < \ldots < j_ i \leq n} x_{j_1}x_{j_2} \ldots x_{j_ i} \]

The existence of $P_ p$ comes from the well known fact that the elementary symmetric functions generate the ring of all symmetric functions over the integers. Another way to characterize $P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ]$ is that we have

\[ \log (1 + c_1 + c_2 + c_3 + \ldots ) = \sum \nolimits _{p \geq 1} (-1)^{p - 1}\frac{P_ p}{p} \]

as formal power series. This is clear by writing $1 + c_1 + c_2 + \ldots = \prod (1 + x_ i)$ and applying the power series for the logarithm function. Expanding the left hand side we get

\begin{align*} & (c_1 + c_2 + \ldots ) - (1/2)(c_1 + c_2 + \ldots )^2 + (1/3)(c_1 + c_2 + \ldots )^3 - \ldots \\ & = c_1 + (c_2 - (1/2)c_1^2) + (c_3 - c_1c_2 + (1/3)c_1^3) + \ldots \end{align*}

In this way we find that

\begin{align*} P_1 & = c_1, \\ P_2 & = c_1^2 - 2c_2, \\ P_3 & = c_1^3 - 3c_1c_2 + 3c_3, \\ P_4 & = c_1^4 - 4c_1^2c_2 + 4c_1c_3 + 2c_2^2 - 4c_4, \end{align*}

and so on. Since the Chern classes of a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ are the elementary symmetric polynomials in the Chern roots $x_ i$, we see that

\[ P_ p(\mathcal{E}) = \sum x_ i^ p \]

For convenience we set $P_0 = r$ in $\mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$ so that $P_0(\mathcal{E}) = r(\mathcal{E})$ as a bivariant class (as in Remarks 42.37.10 and 42.42.5).


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