# Singular          #### D.15.4.43 eulerChProj

Procedure from library `chern.lib` (see chern_lib).

Usage:
eulerChProj(n, r, c); n integer, r polynomial (or integer), c list of polynomials

Return:
polynomial

Purpose:
computes the Euler characteristic of a vector bundle on P_n in terms of its rank and Chern classses

Note:

Example:
 ```LIB "chern.lib"; ring h=0, (r, c(1..3)), ws(0,1,2,3); list l=c(1..3); // the Euler characteristic of a vector bundle on the projective line print( eulerChProj(1, r, l) ); ==> r+c(1) // the Euler characteristic of a vector bundle on the projective plane print( eulerChProj(2, r, l) ); ==> r+3/2*c(1)+1/2*c(1)^2-c(2) // the Euler characteristic of a vector bundle on P_3 print( eulerChProj(3, r, l) ); ==> r+11/6*c(1)+c(1)^2-2*c(2)+1/6*c(1)^3-1/2*c(1)*c(2)+1/2*c(3) // assume now that we have a bundle framed at a subplane of P_3 // this implies c(1)=c(2)=0 l= 0, 0, c(3); // the Euler characteristic is print( eulerChProj(3, r, l) ); ==> r+1/2*c(3) // which implies that c(3) must be even in this case ```

### Misc 