Sturm sequences and number of sign changes of P on ]a; b] : sturm

### 2.25.15 Sturm sequences and number of sign changes of P on ]a; b] : `sturm`

`sturm` takes two or four arguments : P a polynomial expression or P/Q a rational fraction and a variable name or P a polynomial expression, a variable name and two real or complex numbers a and b.

If `sturm` takes two arguments, `sturm` returns the list of the Sturm sequences and multiplicities of the square-free factors of P (or P/Q) (in this case `sturm` behaves like `sturmseq`).

If `sturm` takes four arguments, it behaves like `sturmab` :

- if a and b are reals, `sturm` returns the number of sign changes of P on ]a; b]
- if a or b are complex, `sturm` returns the number of complex roots of P in the rectangle having a and b as opposite vertices.

Input :

`sturm(2*x```^```3+2,x)`

Output :

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

Input :

`sturm((2*x```^```3+2)/(x+2),x)`

Output :

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

Input :

`sturm(x```^```2*(x```^```3+2),x,-2,0)`

Output :

`1`