Resultant
Polynomial expression indicating common roots of two polynomials.
The resultant is a polynomial expression derived from the coefficients of two polynomials. It is equal to zero if and only if the polynomials share a common root or common factor, possibly in a field extension. It is a fundamental tool in number theory, computer algebra, and elimination theory.
- field
- Mathematics
- known_for
- Resultant of polynomials, discriminant, elimination theory
- also_called
- Eliminant (in some older texts)
Lore & Background
The resultant of two univariate polynomials is commonly defined as the determinant of their Sylvester matrix. For polynomials A of degree d and B of degree e, the resultant is computed from a (d+e)×(d+e) matrix formed from their coefficients. The resultant is zero precisely when the polynomials have a common root or common factor.
Reader's Guide
The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. It is a basic tool of computer algebra, built into most computer algebra systems, and is used for cylindrical algebraic decomposition, integration of rational functions, and drawing curves defined by bivariate polynomial equations. The multivariate resultant, or Macaulay's resultant, generalizes the concept to n homogeneous polynomials in n variables and is, with Gröbner bases, one of the main tools of elimination theory.
Did You Know?
- The resultant is also called the eliminant in some older texts.
- The discriminant is essentially the resultant of a polynomial and its derivative.
- The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer.
- The multivariate resultant is also known as Macaulay's resultant.
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