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# SturmSequence -- the Sturm sequence of a rational univariate polynomial

## Synopsis

• Usage:
SturmSequence(f)
• Inputs:
• f, , a rational univariate polynomial
• Outputs:
• a list, the Sturm sequence of f

## Description

This computes the Sturm sequence of the square-free part of a univariate polynomial f.

 i1 : R = QQ[t] o1 = R o1 : PolynomialRing i2 : f = 45 - 39*t - 34*t^2 + 38*t^3 - 11*t^4 + t^5 5 4 3 2 o2 = t - 11t + 38t - 34t - 39t + 45 o2 : R i3 : roots f o3 = {-1, 1, 3, 3, 5} o3 : List i4 : SturmSequence(f) 4 3 2 3 2 104 2 432 232 o4 = {t - 8t + 14t + 8t - 15, 5t - 29t + 27t + 13, ---t - ---t + ---, 25 25 25 ------------------------------------------------------------------------ 3100 5300 194688 ----t - ----, ------, 0} 169 169 24025 o4 : List

• SturmCount -- the number of real roots of a rational univariate polynomial

## Ways to use SturmSequence :

• SturmSequence(RingElement)

## For the programmer

The object SturmSequence is .