Function: polredbest
Section: number_fields
C-Name: polredbest
Prototype: GD0,L,
Help: polredbest(T,{flag=0}): reduction of the polynomial T (gives minimal
polynomials only). If flag=1, gives also elements.
Doc: finds a polynomial with reasonably
small coefficients defining the same number field as $T$.
All $T$ accepted by \tet{nfinit} are also allowed here (e.g. non-monic
polynomials, \kbd{nf}, \kbd{bnf}, \kbd{[T,Z\_K\_basis]}). Contrary to
\tet{polredabs}, this routine runs in polynomial time, but it offers no
guarantee as to the minimality of its result.
This routine computes an LLL-reduced basis for an order in $\Q[X]/(T)$, then
examines small linear combinations of the basis vectors, computing their
characteristic polynomials. It returns the \emph{separable} polynomial $P$ of
smallest discriminant, the one with lexicographically smallest
\kbd{abs(Vec(P))} in case of ties. This is a good candidate for subsequent
number field computations since it guarantees that the denominators of
algebraic integers, when expressed in the power basis, are reasonably small.
With no claim of minimality, though.
It can happen that iterating this functions yields better and better
polynomials, until it stabilizes:
\bprog
? \p5
? P = X^12+8*X^8-50*X^6+16*X^4-3069*X^2+625;
? poldisc(P)*1.
%2 = 1.2622 E55
? P = polredbest(P);
? poldisc(P)*1.
%4 = 2.9012 E51
? P = polredbest(P);
? poldisc(P)*1.
%6 = 8.8704 E44
@eprog\noindent In this example, the initial polynomial $P$ is the one
returned by \tet{polredabs}, and the last one is stable.
If $\fl = 1$: outputs a two-component row vector $[P,a]$, where $P$ is the
default output and \kbd{Mod(a, P)} is a root of the original $T$.
\bprog
? [P,a] = polredbest(x^4 + 8, 1)
%1 = [x^4 + 2, Mod(x^3, x^4 + 2)]
? charpoly(a)
%2 = x^4 + 8
@eprog\noindent In particular, the map $\Q[x]/(T) \to \Q[x]/(P)$,
$x\mapsto \kbd{Mod(a,P)}$ defines an isomorphism of number fields, which can
be computed as
\bprog
subst(lift(Q), 'x, a)
@eprog\noindent if $Q$ is a \typ{POLMOD} modulo $T$; \kbd{b = modreverse(a)}
returns a \typ{POLMOD} giving the inverse of the above map (which should be
useless since $\Q[x]/(P)$ is a priori a better representation for the number
field and its elements).