A note on the discriminant

TL;DR

This paper investigates the order of the discriminant of a polynomial in multiple variables and provides a criterion for the hypersurface to be nonsingular based on the discriminant's properties.

Contribution

It introduces a discriminant criterion for the nonsingularity of hypersurfaces defined by polynomials with coefficients in an algebraically closed field.

Findings

Derived a relation between discriminant order and hypersurface singularity

Established a criterion for nonsingularity based on discriminant non-vanishing

Analyzed the behavior of the discriminant in multivariable polynomial families

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 $P\in K^n$. As application we get a discriminant criterion for the hypersurface F=0 to be nonsingular.