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

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

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

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

In this way we find that

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

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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