Title
On the resultant hypersurface
Document Type
Article
Department or Administrative Unit
Mathematics
Publication Date
2-1990
Abstract
The resultant R(f,g) of two polynomials f and g is an irreducible polynomial such that R(f,g) = 0 if and only if the equations f = 0 and g = 0 have one common root. When g = f′∕p, then D(f) = R(f,g) is called the discriminant of f and the discriminant hypersurface Dp = {f ∈ Cp,D(f) = 0} can be identified to the discriminant of a versal deformation of the simple hypersurface singularity Ap−1 : xp = 0. In particular, the fundamental group π = π1(Cp∖Dp) is the famous braid group and Cp∖Dp in fact a K(π,1) space. Here we prove the following. Theorem. π1(Cp+q∖Rp,q) = Z. As Cp∖Dp can be regarded as a linear section of Cp+q∖Rp,q, this theorem shows that by a nongeneric linear section the fundamental group may change drastically, in contrast with the case of generic section.
Recommended Citation
Choudary, A. D. (1990). On the resultant hypersurface. Pacific Journal of Mathematics, 142(2), 259–263. https://doi.org/10.2140/pjm.1990.142.259
Journal
Pacific Journal of Mathematics
Copyright
Copyright © 1990 by Pacific Journal of Mathematics
Comments
This article was originally published in Pacific Journal of Mathematics. The full-text article from the publisher can be found here.