On the resultant hypersurface

A. D. Raza Choudary

Vol. 142 (1990), No. 2, 259–263

DOI: 10.2140/pjm.1990.142.259

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 D_p = {f ∈ C^p,D(f) = 0} can be identified to the discriminant of a versal deformation of the simple hypersurface singularity A_p−1 : x^p = 0. In particular, the fundamental group π = π₁(C^p∖D_p) is the famous braid group and C^p∖D_p in fact a K(π,1) space.

Here we prove the following.

Theorem. π₁(C^p+q∖R_p,q) = Z.

As C^p∖D_p can be regarded as a linear section of C^p+q∖R_p,q, this theorem shows that by a nongeneric linear section the fundamental group may change drastically, in contrast with the case of generic section.

Mathematical Subject Classification 2000

Primary: 32S50

Milestones

Received: 29 April 1988

Published: 1 April 1990