Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of \(A\) series
A bipolynomial of bidegree \((k,l)\) \(\in \mathbb C [z,w]\) is the restriction of a polynomial \[ h(z,w)=w^k+a_1w^{k-1}+\cdots +a_k+ a_{k+1}z+\cdots +a_{k+l-1}z^{l-1}+z^l \] to the ``hyperbola'' \(wz=\gamma\). The discriminant of the space \(P^{k,l}\) of bipolynomials of bidegree \((k,l)\) is the hypersurface \(\Delta^{k,l}\) of bipolynomials having less than \(k+l\) distinct roots. Similarly, the extended discriminant \(\tilde{\Delta}^{k,l}\) is the union of the discriminant with the hyperplane \(\gamma =0\) of improper bipolynomials. Let \(\phi^{k,l}:P^{k,l}-\Delta^{k,l}\to P^{k+1,l}-\Delta^{k+1,l}\) denote the stabilization map given by adding a point from the edge. In this paper, the author proves that the stabilization map \(\phi^{k,l}\) is a cohomology equivalence up to dimension \([k/2]\) and a similar result holds for the complement \(P^{k,l}-\tilde{\Delta}^{k,l}\), too, where \([k/2]\) denotes the integral part of a real number \(k/2\). This type of results is not so surprising if we recall the results due to G. Segal (see the appendix and the main result in [\textit{G. Segal}, Acta Math. 143, 39-72 (1979; Zbl 0427.55006)]). To prove such results, the author uses a Vassiliev type spectral sequence using the simplicial resolution which was first systematically defined by \textit{V. A. Vassiliev} [Complements of discriminants of smooth maps: topology and applications. Transl. Math. Monographs 98, AMS, Providence (1992; Zbl 0762.55001)]).
scientific article; zbMATH DE number 1940718
scientific article