#### 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.