polrootspadic

Function: polrootspadic
Section: polynomials
C-Name: polrootspadic
Prototype: GGL
Help: polrootspadic(f,p,r): p-adic roots of the polynomial f to precision r.
Doc: vector of $p$-adic roots of the polynomial \var{pol}, given to
 $p$-adic precision $r$; the integer $p$ is assumed to be a prime.
 Multiple roots are
 \emph{not} repeated. Note that this is not the same as the roots in
 $\Z/p^r\Z$, rather it gives approximations in $\Z/p^r\Z$ of the true roots
 living in $\Q_p$:
 \bprog
 ? polrootspadic(x^3 - x^2 + 64, 2, 4)
 %1 = [2^3 + O(2^4), 2^3 + O(2^4), 1 + O(2^4)]~
 ? polrootspadic(x^3 - x^2 + 64, 2, 5)
 %2 = [2^3 + O(2^5), 2^3 + 2^4 + O(2^5), 1 + O(2^5)]~
 @eprog\noindent As the second commands show, the first two roots \emph{are}
 distinct in $\Q_p$, even though they are equal modulo $2^4$.

 More generally, if $T$ is an integral polynomial irreducible
 mod $p$ and $f$ has coefficients in $\Q[t]/(T)$, the argument $p$
 may be replaced by the vector $[T,p]$; we then return the roots of $f$ in
 the unramified extension $\Q_p[t]/(T)$.
 \bprog
 ? polrootspadic(x^3 - x^2 + 64*y, [y^2+y+1,2], 5)
 %3 = [Mod((2^3 + O(2^5))*y + (2^3 + O(2^5)), y^2 + y + 1),
       Mod((2^3 + 2^4 + O(2^5))*y + (2^3 + 2^4 + O(2^5)), y^2 + y + 1),
       Mod(1 + O(2^5), y^2 + y + 1)]~
 @eprog

 If \var{pol} has inexact \typ{PADIC} coefficients, this need not
 well-defined; in this case, the polynomial is first made integral by
 dividing out the $p$-adic content, then lifted to $\Z$ using \tet{truncate}
 coefficientwise. Hence the roots given are approximations of the roots of an
 exact polynomial which is $p$-adically close to the input. To avoid pitfalls,
 we advise to only factor polynomials with exact rational coefficients.