Sylvestermatrix

In algebra, the Sylvester matrix of two polynomials is a special set with the coefficients of the polynomials matrix whose determinant gives the resultant of the polynomials. It is named after the British mathematician James J. Sylvester.

Definition

Be a commutative ring. For two polynomials and from the polynomial ring with

The degree is called the quadratic matrix

The Sylvester matrix and. In the illustration unspecified coefficients are to be understood as zero.

Properties

For is the matrix resulting from the Sylvester matrix by deleting the last row of coefficients, the last lines of coefficients and the last column with the exception of th. The polynomial

Is then the -th Subresultante of and; its leading coefficient

Is the -th Hauptsubresultantenkoeffizient. The - te Hauptsubresultantenkoeffizient

Finally, the resultant of and.

Importance

The Hauptsubresultantenkoeffizienten have an important role as a " yardstick" against the greatest common divisor of polynomials: The degree of two polynomials not equal to 0 over a commutative factorial ring integrity is just the smallest with.

• Commutative Algebra
• Ring theory
• Algebraic Number Theory