Sturm sequences : sturmseq

sturmseq takes as argument, a polynomial expression P or a rationnal fraction P/Q and returns the list of the Sturm sequences of the square-free factors of odd multiplicity of P (or of P/Q). For F a square-free factor of odd multiplicity, the Sturm sequence R₁, R₂,... is made from F, F' by a recurrence relation :

• R₁ is the opposite of the euclidian division remainder of F by F' then,
• R₂ is the opposite of the euclidian division remainder of F' by R₁,
• ...
• and so on untill R_k = 0.

Input :

sturmseq(2*x^3+2)

or

sturmseq(2*y^3+2,y)

Output :

[2,[[1,0,0,1],[3,0,0],-9],1]

The first term gives the content of the numerator (here 2), then the Sturm sequence (in list representation) [x³ +1, 3x², - 9].
Input :

sturmseq((2*x^3+2)/(3*x^2+2),x)

Output :

[2,[[1,0,0,1],[3,0,0],-9],1,[1,[[3,0,2],[6,0],-72]]

The first term gives the content of the numerator (here 2), then the Sturm sequence of the numerator ([[1,0,0,1],[3,0,0],-9]), then the content of the denominator (here 1) and the Sturm sequence of the denominator ([[3,0,2],[6,0],-72]). As expressions, [x³ +1, 3x², - 9] is the Sturm sequence of the numerator and [3x² + 2, 6x, - 72] is the Sturm sequence of the denominator.
Input :

sturmseq((x^3+1)^2,x)

Output :

[1,1]

Indeed F = 1.
Input :

sturmseq(3*(3*x^3+1)/(2*x+2),x)

Output :

[3,[[3,0,0,1],[9,0,0],-81],2,[[1,1],1]]

The first term gives the content of the numerator (here 3),
the second term gives the Sturm sequence of the numerator (here 3x^3+1, 9x^2, -81),
the third term gives the content of the denominator (here 2),
the fourth term gives the Sturm sequence of the denominator (x+1,1).
Warning !!!!
P is defined by its symbolic expression.
Input :
sturmseq([1,0,0,1],x),
Output :
Bad argument type.