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Re: Re: simplification

  • To: mathgroup at smc.vnet.net
  • Subject: [mg121970] Re: Re: simplification
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 7 Oct 2011 04:50:18 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j6e693$kef$1@smc.vnet.net> <201110050801.EAA07025@smc.vnet.net> <360851F3-7690-47A6-BC84-E9521100E6F4@mimuw.edu.pl> <A0A54467-5287-4372-9FA6-F0FD865B77D2@mimuw.edu.pl>

On 6 Oct 2011, at 20:28, Andrzej Kozlowski wrote:

>
> On 6 Oct 2011, at 16:57, Andrzej Kozlowski wrote:
>
>>
>> On 5 Oct 2011, at 10:01, Peter Pein wrote:
>>
>>> Am 04.10.2011 07:40, schrieb dimitris:
>>>> Hello.
>>>>
>>>> Let
>>>>
>>>> o1 = 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + Sqrt[35])]];
>>>> o2 = 1 + Sqrt[3] + Sqrt[5] + Sqrt[7];
>>>>
>>>> o1 is equal to o2.
>>>>
>>>> o1 == o2 // FullSimplify
>>>> True
>>>>
>>>> The question is how to make Mathematica to simplify o1 to o2.
>>>>
>>>> Thanks
>>>> Dimitris
>>>>
>>>
>>> With a lot of luck:
>>>
>>> In[1]:= o1 = 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + 
Sqrt[35])]];
>>> ext = Block[{x, poly = RootReduce[o1][[1]]},
>>> Sqrt[Cases[Union @@ Divisors[Abs[CoefficientList[poly[x], x]]],
>>>   1 | _?PrimeQ, 1]]]
>>> o2 = ((Rest[#1] / First[#1]) . ext & )[
>>>      FindIntegerNullVector[Prepend[ext, -o1]]]
>>>
>>> Out[3]= {1, Sqrt[2], Sqrt[3], Sqrt[5], Sqrt[7], Sqrt[19], 
Sqrt[31]}
>>>
>>> Out[4]= 1 + Sqrt[3] + Sqrt[5] + Sqrt[7]
>>>
>>> :-)
>>>
>>
>> Neat, but from the Mathematical point of view the question was posed 
"the wrong way round" in that o1 is mathematically "simpler" than 1 + 
Sqrt[3] + Sqrt[5] + Sqrt[7], since it is already expressed in terms of 
its minimal polynomial. Hence this is the "natural" or "easy" way to go:
>>
>> ToRadicals[RootReduce[1 + Sqrt[3] + Sqrt[5] + Sqrt[7]]]
>>
>> 1 + Sqrt[15 + 2*Sqrt[35] + 2*Sqrt[6*(6 + Sqrt[35])]]
>>
>> in other words, the algebraic "simplification" in this case is 
exactly the opposite of, what might be called, the visual one.
>>
>>
>> There is no natural or unique way to "decompose" algebraic numbers 
that are already reduced into sums etc, of "simpler" summands or factors 
etc. Of course, if we already know an integer basis for an algebraic 
number field containing an algebraic number, than there are ways of 
expressing it in terms of this basis - and this method is an example.
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
>
> Let me correct myself here since I noticed that I did not write what I 
really meant ;-)
>
> What I mean was that the following is the simplest (algebraically) 
form of this algebraic number:
>
> In[80]:= RootReduce[1 + Sqrt[3] + Sqrt[5] + Sqrt[7]]
>
> Out[80]= Root[
> 1024 + 3584 #1 + 640 #1^2 - 1984 #1^3 - 48 #1^4 + 304 #1^5 -
>   32 #1^6 - 8 #1^7 + #1^8 &, 8]
>
> (the same as above, but, of course, without the ToRadicals). 
Mathematically equivalent statement is:
>
> MinimalPolynomial[RootReduce[1+Sqrt[3]+Sqrt[5]+Sqrt[7]],x]
> 1024+3584 x+640 x^2-1984 x^3-48 x^4+304 x^5-32 x^6-8 x^7+x^8
>
> The only simplification one can really make algorithmically with 
algebraic numbers is what RootReduce does - essentially finding the 
minimal polynomial. Mathematica also has an algorithm for converting 
some root objects to radicals but this is usually does not give the 
visually simplest form. There is no algorithm that will discover the 
simples such form in general (indeed, most algebraic numbers can't be 
expressed in radicals at all). On the other hand, one can find the 
minimal polynomial of any algebraic number: e.g.
>
> In[82]:= MinimalPolynomial[1 + Sqrt[3] + Sqrt[5] + Sqrt[7], x]
> Out[82]= 1024 + 3584*x + 640*x^2 - 1984*x^3 - 48*x^4 + 304*x^5 - 
32*x^6 - 8*x^7 + x^8
>
> In[83]:= 
MinimalPolynomial[1+Sqrt[15+2*Sqrt[35]+2*Sqrt[6*(6+Sqrt[35])]],x]
> Out[83]= 1024+3584 x+640 x^2-1984 x^3-48 x^4+304 x^5-32 x^6-8 
x^7+x^8
>
> You get the same answer which is how Mathematica knowns that these 
numbers are really equal - it does not attempt to transform one into the 
other. When I wrote that
> 1+Sqrt[15+2*Sqrt[35]+2*Sqrt[6*(6+Sqrt[35])] was "algebraically 
simpler" than 1+Sqrt[3]+Sqrt[5]+Sqrt[7]] I meant only that the former is 
Mathematica's radical representation of Root[1024 + 3584 #1 + 640 #1^2 - 
1984 #1^3 - 48 #1^4 + 304 #1^5 - 32 #1^6 - 8 #1^7 + #1^8 &, 8] - which 
is indeed the simplest way (algebraically) to express this algebraic 
number, which, of course, has infinitely many other representations in 
terms of radicals (which Mathematica will not be able, in general, to 
convert into one another but will always be able to find their minimal 
polynomial).
>
> This is not a limitation of Mathematica - it is a limitation of the 
radical representation of algebraic numbers.
>
> Andrzej Kozlowski
>
>
>


Even the above was still not quite accurate. MinimalPolynomial of an 
algebraic number, still does not determine it - obviously

MinimalPolynomial[
 Root[1024 + 3584 #1 + 640 #1^2 - 1984 #1^3 - 48 #1^4 + 304 #1^5 -
    32 #1^6 - 8 #1^7 + #1^8 &, 8], x]

024 + 3584 x + 640 x^2 - 1984 x^3 - 48 x^4 + 304 x^5 -
 32 x^6 - 8 x^7 + x^8

MinimalPolynomial[Root[1024+3584 #1+640 #1^2-1984 #1^3-48 #1^4+304 
#1^5-32 #1^6-8 #1^7+#1^8&,7],x]
1024+3584 x+640 x^2-1984 x^3-48 x^4+304 x^5-32 x^6-8 x^7+x^8

but of course these roots themselves are not the same. However, once you 
know the minimal polynomial of an algebraic number you only need to 
isolate it from the other roots to determine it uniquely - which is 
something that Mathematica does automatically when it numbers the roots. 
So, once you determine the minimal polynomials of two algebraic numbers 
we know that if the polynomials are different so are the numbers, if 
they are the same we use root isolation to distinguish or identify them. 
In any case, the point it that no transformations are performed on 
radicals when doing this sort of thing.




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