[GAP Forum] 1/sqrt(polynomial) not recognized

Laurent Bartholdi laurent.bartholdi at gmail.com
Sat Mar 18 10:13:43 GMT 2006


Hi Marta,
1) most systems, like Maple, can compute Galois groups, in some
  standard format (a set of generating permutations, e.g.)
2) like Dima wrote, it's hopeless to compute for degree above a hundred;
  and usually the limit is much lower
3) the expressions you gave don't produce irreducible polynomials. most
  computer command require an irreducible polynomial.

Here's my sample code, in Maple:

A:=(1/sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - sqrt(x^2-4*x))/2)):
B:=(-1/sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + sqrt(x^2-4*x))/2)):
a := (2 - x + sqrt(x^2 - 4*x))/2:
b := (2 - x - sqrt(x^2 - 4*x))/2:
P := n->A*a^(n-1)+B*b^(n-1):
R := k->(-2+x)^2*(3-5*x+x^2)*P(2*k-2)-(8-14*x+7*x^2-x^3)*P(2*k-3):
L := 'factor(convert(series(R(n),x,2*n+3),polynom))'$n=0..5;
        2              2          2            3      2                     2
(-2 + x) , (3 - 5 x + x ) (-2 + x) , (x - 1) (x  - 8 x  + 17 x - 5) (-2 + x) ,

      6       5       4        3        2                     2
    (x  - 13 x  + 63 x  - 140 x  + 142 x  - 59 x + 7) (-2 + x) ,

      8       7        6        5        4        3        2          
           2
    (x  - 17 x  + 117 x  - 418 x  + 827 x  - 898 x  + 502 x  - 124 x +
9) (-2 + x) ,

              9       8        7        6         5         4        
3         2                       2
    (x - 1) (x  - 20 x  + 167 x  - 753 x  + 1979 x  - 3050 x  + 2635 x
 - 1153 x  + 214 x - 11) (-2 + x)

so i assume you're interested in the "big" factor:
galois(L[2]/(x-2)^2);
                                         "2T1", {"S(2)"}, "-", 2, {"(1 2)"}
galois(L[3]/(x-2)^2/(x-1));
                                        "3T1", {"A(3)"}, "+", 3, {"(1 2 3)"}
galois(L[4]/(x-2)^2);
                      "6T16", {"S(6)"}, "-", 720, {"(3 6)", "(1 6)",
"(2 6)", "(4 6)", "(5 6)"}
galois(L[5]/(x-2)^2);
            "8T50", {"S(8)"}, "-", 40320, {"(4 8)", "(1 8)", "(7 8)",
"(2 8)", "(5 8)", "(6 8)", "(3 8)"}
galois(L[6]/(x-2)^2/(x-1));
       "9T34", {"S(9)"}, "-", 362880, {"(8 9)", "(5 9)", "(6 9)", "(7
9)", "(3 9)", "(4 9)", "(1 9)", "(2 9)"}
this is the limit of Maple's implementation.
Best, Laurent

On 3/18/06, marta asaeda <marta31 at gmail.com> wrote:
> Hello,
>
> I am having a problem: I would like to find galois group for each
> polynomial in a sequence of polynomials parametrized by n. It is given
> by a  recursive formula, so it does involve non-polynomials for
> expression in terms of n. I have this (  I'm mixing mathematica and
> gap notations just for this message):
>
> A[x]:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - Sqrt(x^2
> - 4*x))/2))
> B[x]:=(-1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + Sqrt(x^2
> - 4*x))/2))
>
> a := (2 - x + Sqrt[x^2 - 4x])/2
> b := (2 - x - Sqrt[x^2 - 4x])/2
>
> P[n_, x_] := A[x]a^(n - 1) + B[x]b^(n - 1)
>
> R[k_, x_] :=
>     (-2+x)^2(3-5x+x^2) P[2(k - 1), x] - (8-14x+7x^2-x^3) P[2(k - 1) - 1, x]
>
> R[k,x] is a polynomial for any positive integer k. I would like to
> give the list of galois groups for each k, say, 5<k<100, or 1000, just
> as much as it is doable by gap in a few days. If I just set it at
> night, go to bed, and see 1000 of galois groups spitted out, that will
> be wonderful.  However, it seems gap is having problem dealing with
> 1/Sqrt(x^2 - 4*x). I just tried to teach A, B, a, b one by one, so I
> did like
>
> gap> x:=Indeterminate(Rationals);
> gap> Ax:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x -
> Sqrt(x^2 - 4*x))/2));
>
> then I get error message like
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `Sqrt' on 1 arguments called from
> Error( no_method_found ); called from
> <function>( <arguments> ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
>
>
> Could anyone please tell me what I should do ?
>
> Thank you ~
>
> marta
>
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