Discriminant - Discriminant of A Polynomial Over A Commutative Ring

Discriminant of A Polynomial Over A Commutative Ring

The definition of the discriminant of a polynomial in terms of the resultant may easily be extended to polynomials whose coefficients belong to any commutative ring. However, as the division is not always defined in such a ring, instead of dividing the determinant by the leading coefficient, one substitutes the leading coefficient by 1 in the first column of the determinant. This generalized discriminant has the following property which is fundamental in algebraic geometry.

Let f be a polynomial with coefficients in a commutative ring A and D its discriminant. Let φ be a ring homomorphism of A into a field K and be the polynomial over K obtained by replacing the coefficients of f by their images by φ. Then if and only if either the difference of the degrees of f and is at least 2 or has a multiple root in an algebraic closure of K. The first case may be interpreted by saying that has a multiple root at infinity.

The typical situation where this property is applied is when A is a (univariate or multivariate) polynomial ring over a field k and φ is the substitution of the indeterminates in A by elements of a field extension K of k.

For example, let f be a bivariate polynomial in X and Y with real coefficients, such that f=0 is the implicit equation of a plane algebraic curve. Viewing f as a univariate polynomial in Y with coefficients depending on X, then the discriminant is a polynomial in X whose roots are the X-coordinates of the singular points, of the points with a tangent parallel to the Y-axis and of some of the asymptotes parallel to the Y-axis. In other words the computation of the roots of the Y-discriminant and the X-discriminant allows to compute all remarkable points of the curve, except the inflection points.