I want to take some polynomial f and remove all of it's cyclotomic factors, and then look at the resulting polynomial (say g). I'm aware of polcyclofactors and the current code I have tried is:
c(f)=polcyclofactors(f)
p(f)=prod(i=1,#c(f),c(f)[i])
g(f)=f/p(f)
The issue I have is that polcyclofactors doesn't take into account multiplicity of the cyclotomic factors. For example:
f=3*x^4 + 8*x^3 + 6*x^2 - 1
g(f)
= 3*x^3 + 5*x^2 + x - 1
But
factor(f)
=
[ x + 1 3]
[3*x - 1 1]
Is there any way to be able to nicely include multiple cyclotomic factors of f to divide by? Or will I have to look at factorising f and try and removing cyclotomic factors that way?
The following two suggestions are based on repeating the divisions until no more can be done (they are both very similar).
Suggestion 1:
r(f)={my(c); while(c=polcyclofactors(f); #c, f=f/vecprod(c)); f}
Suggestion 2:
r(f)={my(g=vecprod(polcyclofactors(f))); while(poldegree(g), f=f/g; g=gcd(f,g)); f}
Another suggestion without a loop:
r(f)={my(g=vecprod(polcyclofactors(f))); numerator(f/g^(poldegree(f)))}
Now a version that is probably superior: For each factor valuation can be used to get the required power.
r(f)={f/vecprod([t^valuation(f,t) | t<-polcyclofactors(f)])}