Function: rnfpolredbest
Section: number_fields
C-Name: rnfpolredbest
Prototype: GGD0,L,
Help: rnfpolredbest(nf,pol,{flag=0}): given a pol with coefficients in nf,
 finds a relative polynomial P defining the same field, hopefully simpler
 than pol; flag
 can be 0: default, 1: return [P,a], where a is a root of pol
 2: return an absolute polynomial Pabs, 3:
 return [Pabs, a,b], where a is a root of nf.pol and b is a root of pol.
Doc: relative version of \kbd{polredbest}. Given a polynomial \var{pol}
 with coefficients in $\var{nf}$, finds a simpler relative polynomial $P$
 defining the same field. As opposed to \tet{rnfpolredabs} this function does
 not return a \emph{smallest} (canonical) polynomial with respect to some
 measure, but it does run in polynomial time.

 The binary digits of $\fl$ correspond to $1$: add information to convert
 elements to the new representation, $2$: absolute polynomial, instead of
 relative. More precisely:

 0: default, return $P$

 1: returns $[P,a]$ where $P$ is the default output and $a$,
 a \typ{POLMOD} modulo $P$, is a root of \var{pol}.

 2: returns \var{Pabs}, an absolute, instead of a relative, polynomial.
 Same as but faster than
 \bprog
   rnfequation(nf, rnfpolredbest(nf,pol))
 @eprog

 3: returns $[\var{Pabs},a,b]$, where \var{Pabs} is an absolute polynomial
 as above, $a$, $b$ are \typ{POLMOD} modulo \var{Pabs}, roots of \kbd{nf.pol}
 and \var{pol} respectively.

 \bprog
 ? K = nfinit(y^3-2); pol = x^2 +x*y + y^2;
 ? [P, a] = rnfpolredbest(K,pol,1);
 ? P
 %3 = x^2 - x + Mod(y - 1, y^3 - 2)
 ? a
 %4 = Mod(Mod(2*y^2+3*y+4,y^3-2)*x + Mod(-y^2-2*y-2,y^3-2),
          x^2 - x + Mod(y-1,y^3-2))
 ? subst(K.pol,y,a)
 %5 = 0
 ? [Pabs, a, b] = rnfpolredbest(K,pol,3);
 ? Pabs
 %7 = x^6 - 3*x^5 + 5*x^3 - 3*x + 1
 ? a
 %8 = Mod(-x^2+x+1, x^6-3*x^5+5*x^3-3*x+1)
 ? b
 %9 = Mod(2*x^5-5*x^4-3*x^3+10*x^2+5*x-5, x^6-3*x^5+5*x^3-3*x+1)
 ? subst(K.pol,y,a)
 %10 = 0
 ? substvec(pol,[x,y],[a,b])
 %11 = 0
 @eprog
