Resultant

In mathematics, the resultant is a tool of commutative algebra to two polynomials to test for the existence of common zeros. In extension to multivariate polynomial systems of equations, the resultant can be used to successively eliminate the variables of the system. To this end, the resultant and similar constructions in the course of the 19th century was examined first for systems with symmetries, 1882 by L. Kronecker also for the general case. In modern computer algebra systems resultants and their multidimensional analogues are used to from a predetermined Gröbner basis of the solutions (or their approximations ) to include a system of equations.

Colloquially, a " resultant " is also the result of conflicting decisions or social actions.

Definition

Let and be two polynomials of degree or out, the polynomial ring in one variable over a commutative unitary ring advertised,

The resultant of these two polynomials is the determinant of the Sylvester matrix.

The matrix consists of rows and lines by the coefficients of the coefficient. All in the above matrix is ​​not labeled entries are zero. Sylvester matrix is thus a square matrix having rows and columns.

Properties

The ( transpose of ) Sylvester matrix of the system matrix equation, regarded as a linear equation system in the coefficients of the polynomials cofactor

Do the polynomials and a common factor, the resultant vanishes. For the statement in the other direction is still required that the ring R is a factorial integral domain, ie is without zero divisors and with unique prime factorization. This is always the case when R is a body, for example, the field of rational or real numbers and polynomial rings over it. If these conditions are met, and is so f and g contain a common factor of positive degree.

If the coefficient field an algebraically closed field, as the field of complex numbers, then decompose the polynomials f and g into linear factors

In this case, the resultant can be represented as expressions in the zeros, apply it

By means of Cramer's rule, one can show that there is always polynomials with coefficients A and B in R such that

Applies. The coefficients of A and B result from the last column of the matrix of Sylvester complementary matrix.

Relationship to the Euclidean algorithm

A similar formula is obtained by the extended Euclidean algorithm. In fact, it can be deduced from this an efficient calculation method for the resultant, the Subresultanten process.