Editing Sturm's theorem (section)

==Use of pseudo-remainder sequences==
In [[computer algebra]], the polynomials that are considered have integer coefficients or may be transformed to have integer coefficients. The Sturm sequence of a polynomial with integer coefficients generally contains polynomials whose coefficients are not integers (see above example). 

To avoid computation with [[rational number]]s, a common method is to replace [[Euclidean division of polynomials|Euclidean division]] by [[pseudo-remainder|pseudo-division]] for computing [[polynomial greatest common divisor]]s. This amounts to replacing the remainder sequence of the [[Euclidean algorithm for polynomials|Euclidean algorithm]] by a [[pseudo-remainder sequence]], a pseudo remainder sequence being a sequence <math>p_0, \ldots, p_k</math> of polynomials such that there are constants <math>a_i</math> and <math>b_i</math> such that <math>b_ip_{i+1}</math> is the remainder of the Euclidean division of <math>a_ip_{i-1}</math> by <math>p_i.</math> (The different kinds of pseudo-remainder sequences are defined by the choice of <math>a_i</math> and <math>b_i;</math> typically, <math>a_i</math> is chosen for not introducing denominators during Euclidean division, and <math>b_i</math> is a common divisor of the coefficients of the resulting remainder; see [[Pseudo-remainder sequence]] for details.)
For example, the remainder sequence of the Euclidean algorithm is a pseudo-remainder sequence with <math>a_i=b_i=1</math> for every {{mvar|i}}, and the Sturm sequence of a polynomial is a pseudo-remainder sequence with <math>a_i=1</math> and <math>b_i=-1</math> for every {{mvar|i}}.

Various pseudo-remainder sequences have been designed for computing greatest common divisors of polynomials with integer coefficients without introducing denominators (see [[Pseudo-remainder sequence]]). They can all be made generalized Sturm sequences by choosing the sign of the <math>b_i</math> to be the opposite of the sign of the <math>a_i.</math> This allows the use of Sturm's theorem with pseudo-remainder sequences.