Discriminant in Mathematics

Definition

• The discriminant of a polynomial is a quantity that helps deduce properties of the roots without computing them.
• It is a polynomial function of the coefficients and is used in polynomial factoring, number theory, and algebraic geometry.

Quadratic Polynomial

• For a quadratic polynomial ( ax^2 + bx + c ), the discriminant is ( b^2 - 4ac ).
• Indicates:
  • Zero if the polynomial has a double root.
  • Positive for two distinct real roots.
  • Negative for two distinct complex conjugate roots.

Cubic Polynomial

• Zero if the polynomial has a multiple root.
• Positive for three distinct real roots (real coefficients).
• Negative for one real root and two distinct complex conjugate roots.

General Properties

• For a univariate polynomial, the discriminant is zero if there are multiple roots.
• Positive if the number of non-real roots is a multiple of 4.
• Negative otherwise.

Historical Context

• Term "discriminant" coined in 1851 by James Joseph Sylvester.

Expressions in Terms of Roots

• Involves the square of the Vandermonde polynomial.
• Zero if there are multiple roots.

Higher Degree Polynomials

• Discriminant complexity increases with degree.
• Quartic, quintic, and sextic polynomials have increasingly complex discriminants.

Invariance and Properties

• Discriminant is invariant under projective transformations and ring homomorphisms.
• Product of polynomials affects the discriminant through resultants.

Algebraic Geometry

• Discriminants study algebraic curves and hypersurfaces.
• Helps identify singularities and tangent hyperplanes.

Quadratic Forms

• Discriminant of a quadratic form is the determinant of its associated matrix.
• Invariant under linear transformations.

Real Quadric Surfaces

• Defined by polynomials of degree two in three variables.
• Discriminant helps understand the surface's nature, like curvature and singularities.

Discriminant of an Algebraic Number Field

• Measures size of the ring of integers in a number field.
• Important in analytic formulas and number theory.

Fundamental Discriminants

• Used in studying quadratic fields and binary quadratic forms.
• Criteria for being a fundamental discriminant involves congruence and square-free properties.

Quadratic Number Fields

• Field extension of rationals of degree 2.
• Connection between quadratic fields and forms through discriminants.

Additional References

• Includes references to various academic sources and mathematical entities like OEIS.