resultant
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In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant.
~35 min read
Article
32 sectionsContents
- Notation
- Definition
- Properties
- Characterizing properties
- Zeros
- Invariance by ring homomorphisms
- Invariance under change of variable
- Invariance under change of polynomials
- Generic properties
- Homogeneity
- Elimination property
- Computation
- Application to polynomial systems
- Case of two equations in two unknowns
- General case
- Other applications
- Number theory
- Algebraic geometry
- Symbolic integration
- Computer algebra
- Homogeneous resultant
- Macaulay's resultant
- Resultant of generic homogeneous polynomials
- Properties of the generic Macaulay resultant
- Resultant of polynomials over a field
- Computability
- ''U''-resultant {{anchor|Uresultant}}
- Extension to more polynomials and computation
- See also
- Notes
- References
- External links
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant.
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. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of computer algebra, and is a built-in function of most computer algebra systems. It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation.