Re: Mathematica "Reduce" function

[Denis Simon <denis.simon@unicaen.fr>]

Dear Pedro,

Here is a small gp script that mixes exhaustive search and lifting.
This should be not too bad if the prime number p is small (typically 2 or 3 as in your examples),
and in any cases much better than naive exhaustive search.

Idea of the algo : exhaustive search mod p, then exhaustive lift mod p^2, mod p^3...

Sincerely,
Denis SIMON.


p = 3;  \\ your prime number
k = 2;  \\ the exponent of p
eqns = [ x^2 + 3*y^2 - 4, 3*x^3 - 4*y^2 + x*y - 11 ];   \\ list of equations

{
vars = variables(eqns);                                 \\ list of variables
n = #vars;
sol_modpi = [vector(n,j,0)]; \\ this vector contains the solutions mod p^i, initialiszed for i=0
pi = 1;                      \\ = p^i
pi1= p;                      \\ = p^(i+1)

for( i = 0, k-1,
  sol_modpi1 = [];             \\ solution mod p^(i+1)
  for( j = 1, #sol_modpi,
    sol = sol_modpi[j];      \\ select a solution mod p^i
    forvec( X = vector(n,j,[0,p-1]),
      try = sol + pi*X;
      if(substvec(eqns,vars,try)%pi1==0,sol_modpi1 = concat(sol_modpi1,[try]));
    );
  );
  print("nb of sol mod ",p,"^",i+1," = ",#sol_modpi1);
  sol_modpi = sol_modpi1;
  pi = pi1;
  pi1*=p;
);
}


----- Mail original -----
> De: "Bill Allombert" <Bill.Allombert@math.u-bordeaux.fr>
> À: pari-users@pari.math.u-bordeaux.fr
> Envoyé: Jeudi 2 Mars 2017 11:26:35
> Objet: Re: Mathematica "Reduce" function
> 
> On Thu, Mar 02, 2017 at 10:00:48AM +0100, Pedro Fortuny Ayuso wrote:
> > Thanks to all.
> > 
> > My specific problem is trying to solve equations like
> > 
> > 6x^2 + 12y^2 +20z^2 = 0
> > 
> > over Z/(2^k)Z. That is, finding the points of that surface
> > over that ring.
> 
> Solutions of homogenous degree-2 equation in three variables can be
> parametrized as soon as one solution is known using qfparam:
> For example [0,1,1] is a (primitive) solution mod 2^5 so set
> 
> ? M=qfparam(matdiagonal([6,12,20]),[0,1,1])
> %9 = [0,-20,0;3,0,10;3,0,-10]
> ? v = y^2 * M*[1,x/y,(x/y)^2]~
> %10 = [-20*y*x,10*x^2+3*y^2,-10*x^2+3*y^2]~
> 
> so for all x,y, (-20*y*x,10*x^2+3*y^2,-10*x^2+3*y^2) is a solution mod
> 2^5:
> 
> ? content(6*(-20*y*x)^2+12*(10*x^2+3*y^2)^2+20*(-10*x^2+3*y^2)^2)
> %12 = 32
> 
> > Bill's reply of counting
> > 
> > length([[x,y,z]|x<-[0..2^k-1];y<-[0..2^k-1];z<-[0..2^k-1],6*x^2+12*y^2+20*z^2==0])
> 
> You are missing a reduction mod 2^k at the end.
> 
> > is the fastest but it ***looks like*** a lot slower than
> > Mathematica (but please notice I am working on a system
> > with pari/gp and my colleague on a different one with Mathematica,
> > so that it may have nothing to do with pari/Mathematica).
> 
> This is quite possible, I do not know what Mathematica is doing.
> what you can do is to check whether (6*x^2+12*y^2)/-20 is a square
> instead of iterating over z:
> 
> [[x,y,z]|x<-2*[0..2^(k-1)-1];y<-[0..2^k-1],issquare((6*x^2+12*y^2)/-20*Mod(1,2^k),&z)]
> 
> Cheers,
> Bill.
> 
>