# 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 rational univariate polynomial
• g, , a rational univariate polynomial
• a, , (optional) the lower bound of the interval
• b, , (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

## 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 .