# A “well known interpolation formula”

In the Atiyah-Bott paper on their Lefschetz theorem for elliptic complexes, they give a very nice elementary example of the Lefschetz theorem for the Dolbeault complex, by considering the automorphism of  ${\mathbb C}{\mathbb P}^n$ given by

$[z_0,\ldots, z_n] \mapsto [\gamma_0z_0, \ldots, \gamma_nz_n ]$

in homogeneous coordinates, where the $$\gamma_i$$ are distinct and nonzero complex numbers.  This has $$(n+1)$$ simple fixed points and applying the holomorphic Lefschetz theorem gives

$1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}.$

This is Example 2 on p. 460 of the second Atiyah-Bott paper.  They go on to describe this as a “well known interpolation formula”. I’m teaching this material in my elliptic topology class at the moment.  I had a vague memory of working out how this formula related to Lagrange polynomial interpolation for my “red book” – but when I checked I found that I had just put it in an exercise with no hints (and, to make matters worse, I had confused Lagrange with Legendre).  Anyhow, here is how it works.

Let $$x_1,\ldots,x_n$$ be $$n$$ distinct numbers.   The $$j$$’th Lagrange primary factor associated to these points is the polynomial

$L_j(x) = \prod_{k\neq j} \left (\frac{x-x_k}{x_j-x_k}\right).$

By construction, $$L_j(x_i) = \delta_{ij}$$.  This gives the Lagrange interpolation formula

$p(x) = \sum_{j=1}^n a_j L_j(x)$

for the unique polynomial $$p$$ of degree at most $$n-1$$ that has $$p(x_k)=a_k$$ for $$k=1,\ldots,n$$.

Consider in particular the case $$a_k = (x_k)^n$$, so we are looking for the unique polynomial of degree at most $$n-1$$ that coincides with $$x^n$$ at the $$n$$ given points.  On the one hand we have the Lagrange formula; on the other hand, it is obvious that

$p(x) = x^n – (x-x_1)\cdots (x-x_n)$

is a polynomial of degree at most $$n-1$$ with the required properties.   By uniqueness then

$x^n – \prod_{k=1}^n (x-x_k) = \sum_{j=1}^n\left( x_j^n \prod_{k\neq j} \frac{x-x_k}{x_j-x_k}\right).$

Dividing through by $$\prod_{k=1}^n (x-x_k)$$ and then  putting $$x_j=\gamma_j$$ for $$j=1,\ldots,n$$, and $$x=\gamma_0$$, we get Atiyah and Bott’s result.

### Reference

Atiyah, M. F., and R. Bott. 1968. “A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications.” Annals of Mathematics 88 (3): 451–91. doi:10.2307/1970721.