Re: FindInstance what inspite ?

```On 3 Dec 2007, at 19:39, 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 {}
> Who know how I can realize that procedure in Mathematica ?
>
> Best wishes
> Artur
>
>
>

First of all your answer is incorrect. You can check it with
Mathematica:

I FullSimplify[
Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 2] +
Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 3] ==
{-2, -3, -2, -1, 2} .
Table[Root[#1^5 - #1^3 - 2*#1^2 - 2*#1 - 1 & ,
1]^i, {i, 0, 4}]]
False

The correct answer is {2, 2, 2, 1, -2} or, if you prefer
{-2,-2,-2,-1,2}, whic differs from yours in the second place.

Neither FindInstance not Reduce or any other general algorithm based
on polynomial algebra (or algebraic geometry) will work because they
all work over the real or compelex number fields and you are looking
for integer solutions. But in Mathematica 6 you can find this answer
as follows:

ToNumberField[Root[-1 - 2*#1 - 2*#1^2 - #1^3 +
#1^5 & , 2] +
Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 3],
Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 1]]

AlgebraicNumber[Root[#1^5 - #1^3 - 2*#1^2 - 2*#1 -
1 & , 1], {2, 2, 2, 1, -2}]

You can check that is correct:

FullSimplify[
Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 2] +
Root[-1 - 2*#1 - 2*#1^2 - #1^3 + #1^5 & , 3] ==
{2, 2, 2, 1, -2} .
Table[Root[#1^5 - #1^3 - 2*#1^2 - 2*#1 - 1 & ,
1]^i, {i, 0, 4}]]
True

Best regards
Andrzej Kozlowski

```

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