Resultant
Polynomial expression indicating common roots of two polynomials.
The resultant is a polynomial expression constructed from the coefficients of two polynomials. Its defining property is that it equals zero precisely when the two polynomials possess a common root, which may lie in an extension of the coefficient field, or equivalently, when they share a common factor. In older texts, it was sometimes called the eliminant. The resultant is widely applied in number theory, often indirectly through the discriminant, which is essentially the resultant of a polynomial and its derivative. It is a fundamental tool in computer algebra, efficiently computable for polynomials with rational or polynomial coefficients, and is a built-in function in most computer algebra systems. Applications include cylindrical algebraic decomposition, integration of rational functions, and plotting curves defined by bivariate polynomial equations. A generalization, the multivariate resultant or Macaulay's resultant, extends the concept to systems of homogeneous polynomials in several variables and is a key technique in elimination theory alongside Gröbner bases. The resultant is commonly defined as the determinant of the Sylvester matrix, a linear map between vector spaces of polynomials of bounded degree. If the coefficients belong to an integral domain, the resultant can be expressed as a product involving the roots of the polynomials in an algebraically closed field. Key properties include that the resultant is zero if and only if the polynomials have a common divisor of positive degree over the field of fractions, and that there exist polynomials of lower degree satisfying a Bézout-like identity. The resultant is also invariant under certain ring homomorphisms, provided degrees are preserved, and under linear and projective changes of the variable.
- 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, possibly in a field extension. In older texts, it is sometimes called the eliminant. The resultant can also be expressed as a product involving the roots of the polynomials, taken with multiplicities, in any algebraically closed field containing the coefficient domain. It satisfies several characterizing properties: it is invariant under ring homomorphisms that preserve the degrees of the polynomials, and it is also invariant under taking reciprocal polynomials. If the leading coefficient of one polynomial is preserved under a ring homomorphism, the resultant of the specialized polynomials equals the specialization of the resultant. This property is fundamental for applications such as cylindrical algebraic decomposition. The resultant is a basic tool in computer algebra, used for integration of rational functions and for drawing curves defined by bivariate polynomial equations. A generalization, the multivariate resultant (or Macaulay’s resultant), extends the concept to n homogeneous polynomials in n variables and is a main tool of elimination theory alongside Gröbner bases.
Reader's Guide
The resultant is a polynomial expression derived from the coefficients of two polynomials, defined as the determinant of their Sylvester matrix. Its fundamental property is that it equals zero precisely when the two polynomials share a common root or common factor, even if that root exists only in an extended field. This makes it a central tool in elimination theory, where it is used alongside Gröbner bases. The resultant is invariant under certain transformations, such as linear and projective changes of the variable, and its behavior under ring homomorphisms allows for efficient computation, for instance by evaluating it modulo several primes and reconstructing the result via the Chinese remainder theorem. It also satisfies a generalization of Bézout's identity, meaning the resultant lies within the ideal generated by the two polynomials. In number theory, the resultant appears directly or through the discriminant, which is the resultant of a polynomial and its derivative. In computer algebra, it is a built-in function used for tasks like cylindrical algebraic decomposition, integrating rational functions, and plotting curves defined by bivariate polynomial equations. The multivariate resultant, introduced by Macaulay, extends the concept to systems of n homogeneous polynomials in n variables.
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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