Sturm sequences : sturmseq

### 5.28.17 Sturm sequences : sturmseq

sturmseq takes as argument, a polynomial expression P or a rational 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 euclidean division remainder of F by F′ then,
- R2 is the opposite of the euclidean division remainder of F′ by R1,
- ...
- and so on until Rk=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.