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sylvesterSequence -- the Sylvester sequence of two univariate polynomials with rational coefficients

Description

This computes the reduced Sylvester sequence of two univariate polynomials with rational coefficients f and g in the same ring. This begins with the Sylvester sequence $(f_{0},f_{1},\dots,f_{k})$, where $f_{0} = f, f_{1} = f'\cdot g$ and for $i\geq 1, f_{i+1} = -1\cdot$remainder $(f_{i-1},f_{i})$. The last nonzero remainder $f_{k}$ is a greatest common divisor of $f$ and $g$. The reduced Sylvester sequence is obtained by dividing each term of the Sylvester sequence by $f_{k}$.

i1 : R = QQ[t]

o1 = R

o1 : PolynomialRing
i2 : f = (t + 1)*(t + 2)

      2
o2 = t  + 3t + 2

o2 : R
i3 : g = t + 2

o3 = t + 2

o3 : R
i4 : sylvesterSequence(f,g)

                     1
o4 = {t + 1, 2t + 3, -, 0}
                     2

o4 : List

See also

Ways to use sylvesterSequence:

  • sylvesterSequence(RingElement,RingElement)

For the programmer

The object sylvesterSequence is a method function.


The source of this document is in RealRoots.m2:773:0.