Editing Discriminant (section)

==Use in algebraic geometry==

The typical use of discriminants in [[algebraic geometry]] is for studying plane [[algebraic curve]]s, and more generally [[Hypersurface|algebraic hypersurface]]s. Let {{math|''V''}} be such a curve or hypersurface; {{math|''V''}} is defined as the zero set of a [[multivariate polynomial]]. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface {{math|''W''}} in the space of the other indeterminates. The points of {{math|''W''}} are exactly the projection of the points of {{math|''V''}} (including the [[points at infinity]]), which either are singular or have a [[tangent space|tangent hyperplane]] that is parallel to the axis of the selected indeterminate.

For example, let {{mvar|f}} be a bivariate polynomial in {{mvar|X}} and {{mvar|Y}} with real coefficients, so that&nbsp;{{math|1=''f'' &thinsp;= 0}} is the implicit equation of a real plane [[algebraic curve]]. Viewing {{mvar|f}} as a univariate polynomial in {{mvar|Y}} with coefficients depending on {{mvar|X}}, then the discriminant is a polynomial in {{mvar|X}} whose roots are the {{mvar|X}}-coordinates of the singular points, of the points with a tangent parallel to the {{mvar|Y}}-axis and of some of the asymptotes parallel to the {{mvar|Y}}-axis. In other words, the computation of the roots of the {{mvar|Y}}-discriminant and the {{mvar|X}}-discriminant allows one to compute all of the remarkable points of the curve, except the [[inflection point]]s.