A Note on the Discriminant

Abstract

Let F(X, Y) = Y ^d  + a ₁(X)Y ^d−1 + … +a _d (X) be a polynomial in n + 1 variables (X, Y) = (X ₁,…, X _n , Y) with coefficients in an algebraically closed field 𝕂. Assuming that the discriminant D(X) = disc _Y F(X, Y) is nonzero we investigate the order ord _P D for P ∈ 𝕂 ⁿ . As application we get a discriminant criterion for the hypersurface F = 0 to be nonsingular.

Key Words:

• Discriminant of a polynomial
• Nonsingular hypersurface

2010 Mathematics Subject Classification:

• 11C08
• 13B25

Notes

Communicated by J.-T. Yu.