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Re: Solve : missing elims in the Mathematica 8 version

  • To: mathgroup at smc.vnet.net
  • Subject: [mg115054] Re: Solve : missing elims in the Mathematica 8 version
  • From: Adam Strzebonski <adams at wolfram.com>
  • Date: Wed, 29 Dec 2010 06:00:00 -0500 (EST)

Andrzej Kozlowski wrote:
> On 28 Dec 2010, at 12:49, W. Deinhard wrote:
> 
>> ?Hi, 
>> in Mathematica 7 Solve allowed to specify the variables I wanted to eliminate.
>> How can I do that in Mathtematica 8 ?
>>
>> The syntax is no longer 
>>
>> Solve[eqns,vars,elims]
>>
>> but 
>>
>> Solve[expr,vars,dom]
>>
>> Thanks , bye Walter.
>>
> 
> First, the old syntax still works but now it is undocumented (?):
> 
> In[33]:= Solve[{1 == x^5 + y^5, a - b == x + y, b + a == x*y}, {x, 
>   y}, {a, b}]
> 
> Solve::svars:Equations may not give solutions for all "solve" variables. >>
> 
> {{y -> (1 - x^5)^(1/
>       5)}, {y -> (-(-1)^(1/5))*(1 - x^5)^(1/5)}, 
>    {y -> (-1)^(2/5)*(1 - x^5)^(1/5)}, 
>    {y -> (-(-1)^(3/5))*(1 - x^5)^(1/5)}, 
>    {y -> (-1)^(4/5)*(1 - x^5)^(1/5)}}
> 
> This, in fact, is the same answer as the one you get if you explicitly use Eliminate:
> 
> Solve[
>  Eliminate[{1 == x^5 + y^5, a - b == x + y, b + a == x*y}, {a, 
>    b}], {x, y}]
> 
> Solve::svars:Equations may not give solutions for all "solve" variables. >>
> 
>  {{y -> (1 - x^5)^(1/
>       5)}, {y -> (-(-1)^(1/5))*(1 - x^5)^(1/5)}, 
>    {y -> (-1)^(2/5)*(1 - x^5)^(1/5)}, 
>    {y -> (-(-1)^(3/5))*(1 - x^5)^(1/5)}, 
>    {y -> (-1)^(4/5)*(1 - x^5)^(1/5)}}
> 
> I am not sure if the lack of documentation for the former usage is an oversight or it is now deprecated. 
> 
> Andrzej Kozlowski

The new Solve syntax allows use of arbitrary combinations
of quantifiers. "Elimination variables" are a special case,
namely

Solve[eqns, vars, elims]

is equivalent to

Solve[Exists[elims, eqns], vars]

In[1]:= Solve[Exists[{a, b}, 1 == x^5 + y^5 && a - b == x + y &&
         b + a == x*y], y]

                      5 1/5               1/5       5 1/5
Out[1]= {{y -> (1 - x )   }, {y -> -((-1)    (1 - x )   )},

                2/5       5 1/5               3/5       5 1/5
 >    {y -> (-1)    (1 - x )   }, {y -> -((-1)    (1 - x )   )},

                4/5       5 1/5
 >    {y -> (-1)    (1 - x )   }}

Note, that the last argument of a quantifier needs to be a Boolean
formula, so one should use

eq1 && eq2 && ...

rather than

{eq1, eq2, ...}

The old syntax still works, but is now deprecated because of
a possible confusion with the domain argument.

In[2]:= Solve[x^4==1, x, Reals]

Out[2]= {{x -> -1}, {x -> 1}}

In[3]:= Solve[x^4==1, x, Real]

Solve::bdomv: Warning: Real is not a valid domain specification. Mathematica
      is assuming it is a variable to eliminate.

Out[3]= {{x -> -1}, {x -> -I}, {x -> I}, {x -> 1}}

Best regards,

Adam Strzebonski
Wolfram Research


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