This property is fundamental, for example, for cylindrical algebraic decomposition. This means that the property of the resultant being zero is invariant under linear and projective changes of the variable. These properties imply that in the Euclidean algorithm for polynomials, and all its variants (pseudo-remainder sequences), the resultant of two successive remainders (or pseudo-remainders) differs from the resultant of the initial polynomials by a factor which is easy to compute. Conversely, this allows one to deduce the resultant of the initial polynomials from the value of the last remainder or pseudo-remainder. This is the starting idea of the subresultant-pseudo-remainder-sequence algorithm, which uses the above formulae for getting subresultant polynomials as pseudo-remainders, and the resultant as the last nonzero pseudo-remainder (provided that the resultant is not zero). This algorithm works for polynomials over the integers or, more generally, over an integral domain, without any division other than exact divisions (that is, without involving fractions). It involves formula_55 arithmetic operations, while the computation of the determinant of the Sylvester matrix with standard algorithms requires formula_56 arithmetic operations. In this section, we consider two polynomials and whose coefficients are distinct indeterminates. Let be the polynomial ring over the integers defined by these indeterminates. The resultant formula_1 is often called the generic resultant for the degrees and . It has the following properties. The generic resultant for the degrees and is homogeneous in various ways. More precisely:  Let formula_73 be the ideal generated by two polynomials and in a polynomial ring formula_74 where is itself a polynomial ring over a field. Then: An example where in the latter property is formula_79 formula_80 (the unit circle), and . In this case, formula_81 and formula_82 This example has been chosen for having a prime ideal formula_83 For another field of coefficients and another constant term in , one has also , but the ideal may be non-prime. Theoretically, the resultant could be computed by using the formula expressing it as a product of roots differences. However, as the roots may generally not be computed exactly, such an algorithm would be inefficient and numerically unstable. As the resultant is a symmetric function of the roots of each polynomial, it could also be computed by using the fundamental theorem of symmetric polynomials, but this would be highly inefficient.