%0 Journal Article %T Gaudin functions, and Euler-Poincar¨¦ characteristics %A Alain Lascoux %J Mathematics %D 2007 %I arXiv %X Given two positive integers n,r, we define the Gaudin function of level r to be quotient of the numerator of the determinant det(1/ ((x_i-y_j)(x_i-ty_j) ... (x_i-t^r y_j)), i,j=1..n, by the two Vandermonde in x and y. We show that it can be characterized by specializing the x-variables into the y-variables, multiplied by powers of t. This allows us to obtain the Gaudin function of level 1 (due to Korepin and Izergin) as the image of a resultant under the the Euler-Poincar\'e characteristics of the flag manifold. As a corollary, we recover a result of Warnaar about the generating function of Macdonald polynomials. %U http://arxiv.org/abs/0709.1635v1