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.



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.

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