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Re: FindInstance what inspite ?


Artur wrote:
> Who have idea what function uses inspite FindInstance in procedure?
> \!\(FindInstance[Chop[N[Root[\(-1\) - 2\ #1 - 2\ #1\^2 - #1\^3 + #1\^5 
> &, 2] \
> + Root[\(-1\) - 2\ #1 - 2\ #1\^2 - #1\^3 + #1\^5 &, 3], 500]] == a + b\
>                 Root[\(-1\) - 2\ #1 - 2\ #1\^2 - #1\^3 + #1\^5 &,
>                      1] + c\ Root[\(-1\) - 2\ #1 - 2\ #1\^2 - #1\^3 + 
> #1\^5 \
> &, 1]^2 + d\
>             Root[\(-1\) - 2\ #1 - 2\ #1\^2 - #1\^3 + #1\^5 &, 1]^3 + e\
>                     Root[\(-1\) - 2\ #1 - 2\ #1\^2 - #1\^3 + #1\^5 &,
>                    1]^4 && a != 0, {a, b, c, d, e}, Integers]\)
> And anser is empty set {}
> Good answer  is {a,b,c,d,e}={-2,-3,-2,-1,2}
> Who know how I can realize that procedure in Mathematica ?
> 
> Best wishes
> Artur

One can attempt such problems directly using lattice reduction. The idea 
is to form a vector consisting of your target value and "basis" values 
(the zeroeth through fourth powers of a certain algebraic number, in 
your example). Multiply by a power of 10 raised to the precision you 
have in mind, and round off to get integers. Augment on the right with 
an identity matrix. Reduce this lattice, look for a small vector with 
element in the column corresponding to the input value equal to +-1 (so 
we know we obtained the value itself, and not a nontrivial multiple 
thereof). This last can be relaxed if you are willing to allow rationals 
(with small denominators, say) as coefficients.

So here is code to do all this.

minimalPolynomialInRoot[val_Real, alg_Root, deg_Integer] := Module[
   {vec, prec=Floor[Precision[val]], lat, redlat, mults},
   vec = Round[10^prec*Append[alg^Range[0,deg],-val]];
   lat = Transpose[Prepend[IdentityMatrix[deg+2],vec]];
   redlat = LatticeReduce[lat];
   mults = First[redlat];
   If [Abs[Last[mults]]==1,
     Take[mults,{2,-2}] / Last[mults],
     $Failed]
   ]

Your example:

val = Re[N[Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 2, 0] +
      Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 3, 0], 500]];
alg = Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 1];

In[65]:= multipliers = minimalPolynomialInRoot[val, alg, 4]
Out[65]= {2, 2, 2, 1, -2}

Check:

In[66]:= multipliers.alg^Range[0,deg] - val
               -500
Out[66]= 0. 10


Daniel Lichtblau
WOlfram Research






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