1Function: rnfpolredabs 2Section: number_fields 3C-Name: rnfpolredabs 4Prototype: GGD0,L, 5Help: rnfpolredabs(nf,pol,{flag=0}): given an irreducible pol with coefficients 6 in nf, finds a canonical relative polynomial defining the same field. 7 Binary digits of flag mean: 1: return also the element whose characteristic 8 polynomial is the given polynomial, 2: return an absolute polynomial, 9 16: partial reduction. 10Doc: Relative version of \kbd{polredabs}. Given an irreducible monic polynomial 11 \var{pol} with coefficients in the maximal order of $\var{nf}$, finds a 12 canonical relative 13 polynomial defining the same field, hopefully with small coefficients. 14 Note that the equation is only canonical for a fixed \var{nf}, using a 15 different defining polynomial in the \var{nf} structure will produce a 16 different relative equation. 17 18 The binary digits of $\fl$ correspond to $1$: add information to convert 19 elements to the new representation, $2$: absolute polynomial, instead of 20 relative, $16$: possibly use a suborder of the maximal order. More precisely: 21 22 0: default, return $P$ 23 24 1: returns $[P,a]$ where $P$ is the default output and $a$, 25 a \typ{POLMOD} modulo $P$, is a root of \var{pol}. 26 27 2: returns \var{Pabs}, an absolute, instead of a relative, polynomial. 28 This polynomial is canonical and does not depend on the \var{nf} structure. 29 Same as but faster than 30 \bprog 31 polredabs(rnfequation(nf, pol)) 32 @eprog 33 34 3: returns $[\var{Pabs},a,b]$, where \var{Pabs} is an absolute polynomial 35 as above, $a$, $b$ are \typ{POLMOD} modulo \var{Pabs}, roots of \kbd{nf.pol} 36 and \var{pol} respectively. 37 38 16: (OBSOLETE) possibly use a suborder of the maximal order. This makes 39 \kbd{rnfpolredabs} behave as \kbd{rnfpolredbest}. Just use the latter. 40 41 \misctitle{Warning} The complexity of \kbd{rnfpolredabs} 42 is exponential in the absolute degree. The function \tet{rnfpolredbest} runs 43 in polynomial time, and tends to return polynomials with smaller 44 discriminants. It also supports polynomials with arbitrary coefficients in 45 \var{nf}, neither integral nor necessarily monic. 46