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SylvesterCount -- the difference in variations of the Sylvester sequence of two rational univariate polynomials

Synopsis

Usage:

SylvesterCount(f,g,a,b)

SylvesterCount(f,g)
Inputs:
- f, a ring element, a rational univariate polynomial
- g, a ring element, a rational univariate polynomial
- a, a real number, (optional) the lower bound of the interval
- b, a real number, (optional) the upper bound of the interval
Optional inputs:
- Multiplicity => ..., default value false, option for computing roots with multiplicity
Outputs:
- an integer, the difference between the number of roots of f in the interval $(a,b]$ where g is positive and where g is negative

Description

This computes the difference in variations of the Sylvester sequence of f and f'g on the interval $(a,b]$.

i1 : R = QQ[t]

o1 = R

o1 : PolynomialRing

i2 : f = (t - 2)*(t - 1)*(t + 3)

      3
o2 = t  - 7t + 6

o2 : R

i3 : g = t + 1

o3 = t + 1

o3 : R

i4 : a = -5

o4 = -5

i5 : b = 4

o5 = 4

i6 : SylvesterCount(f,g,a,b)

o6 = 1

See also

SylvesterSequence -- the Sylvester sequence of two rational univariate polynomials

Ways to use SylvesterCount:

SylvesterCount(RingElement,RingElement)
SylvesterCount(RingElement,RingElement,InfiniteNumber,InfiniteNumber)
SylvesterCount(RingElement,RingElement,InfiniteNumber,QQ)
SylvesterCount(RingElement,RingElement,InfiniteNumber,RR)
SylvesterCount(RingElement,RingElement,InfiniteNumber,ZZ)
SylvesterCount(RingElement,RingElement,QQ,InfiniteNumber)
SylvesterCount(RingElement,RingElement,QQ,QQ)
SylvesterCount(RingElement,RingElement,QQ,RR)
SylvesterCount(RingElement,RingElement,QQ,ZZ)
SylvesterCount(RingElement,RingElement,RR,InfiniteNumber)
SylvesterCount(RingElement,RingElement,RR,QQ)
SylvesterCount(RingElement,RingElement,RR,RR)
SylvesterCount(RingElement,RingElement,RR,ZZ)
SylvesterCount(RingElement,RingElement,ZZ,InfiniteNumber)
SylvesterCount(RingElement,RingElement,ZZ,QQ)
SylvesterCount(RingElement,RingElement,ZZ,RR)
SylvesterCount(RingElement,RingElement,ZZ,ZZ)

For the programmer

The object SylvesterCount is a method function with options.