- let be polynomials with coefficients from the
domain
- the domain can be e.g., integers or polynomials
in other variables

- the Sylvester matrix of polynomials is the matrix

- where rows are constructed from the coefficients
of the polynomial and rows from
the coefficients of the polynomial
- the resultant of the polynomials and with respect to the variable , denoted by
, is the determinant of the Sylvester
matrix
- using the special structure of the Sylvester matrix, we can
calculate the resultant by the following recursive algorithm
- Algorithm
res := RES(x, a, b) := [a, b are polynomials algorithms used: rem(a, b) - remainder after dividing a by b deg(a) - degree of the polynomial a lcof(a) - leading coefficient of the polynomial a, i.e., the coefficient of x^deg(a)] 1. n := deg(a) m := deg(b) 2. if n > m then res :=(-1)^(n m) RES(x, b, a) else lc := lcof(a) if n = 0 then res := lc^m else r := rem(b, a) if r = 0 then res := 0 else p := deg(r) res := lc^(m - p) RES(x, a, r) fi fi fi

- this algorithm gradually replaces rows in the Sylvester matrix
derived from the coefficients of the polynomial by
rows derived from the coefficients of the remainder ; these new rows are computed as linear
combinations of the rows of the original matrix
- after the above replacement, the matrix has only zeroes below the diagonal in its first columns while the diagonal has the values ; the matrix block consisting of the last rows and columns is the Sylvester matrix of the polynomials ; thus, in general case of the algorithm, we can use the formula

Richard Liska