Re: FullSimplify with Assumptions

• To: mathgroup at smc.vnet.net
• Subject: [mg53359] Re: [mg53339] FullSimplify with Assumptions
• Date: Sat, 8 Jan 2005 02:39:26 -0500 (EST)
• References: <200501070300.WAA10775@smc.vnet.net> <A3EDD934-606A-11D9-9BFB-000A95B4967A@mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com

```Andrzej Kozlowski wrote:
> *This message was transferred with a trial version of CommuniGate(tm) Pro*
> On 7 Jan 2005, at 12:00, Goyder Dr HGD wrote:
>
>> In the examples below I would expect FullSimplify to give L.
>> However, I get results that depend on the symbols I use.
>> It would appear that symbols x and y are treated differently.
>>
>> How can I force FullSimplify to use the LeafCount as the
>> ComplexityFunction?
>>
>>
>> In[14]:= r1 = FullSimplify[(L - L*y^2)/x^2, {-1 + x^2 + y^2 == 0}]
>>
>> Out[14]= (L - L*y^2)/x^2
>>
>> In[15]:= r2 = FullSimplify[(L - L*x^2)/y^2, {-1 + x^2 + y^2 == 0}]
>>
>> Out[15]= L
>>
>> In[16]:= LeafCount[r1]
>>
>> Out[16]= 12
>>
>> In[17]:= LeafCount[r2]
>>
>> Out[17]= 1
>>
>> In[18]:= \$Version
>>
>> Out[18]= "5.1 for Microsoft Windows (October 25, 2004)"
>>
>> Thanks for any comment
>>
>> Hugh Goyder
>>
> I would speculate that the reason has something to do with FullSimplify
> using PolynomialReduce or related functions, whose outcome depends on
> the ordering of the variables. Compare for example:
>
>
> Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2},
>    {x, y}]]
>
>
> L/x^2 - (L*y^2)/x^2
>
> with
>
>
> Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2},
>    {y, x}]]
>
> L
>
> Note also that if you do not include explicitly the variables you will get:
>
>
> Last[PolynomialReduce[(L - L*y^2)/x^2, {-1 + x^2 + y^2}]]
>
>
> L/x^2 - (L*y^2)/x^2
>
>
> I suspect this isn't a bug in FullSimplify. To remedy it FullSimplify
> would need to use all possible orderings of variables when applying
> algebraic functions whose output depends on variable order, and that is
> obviously just not a reasonable option in terms of performance. Also,
> Simplify and FullSimplify are not really optimized for conditions of the
> form  something == something else; I find it a little surprising that it
> works as well as it does. One should really deal with such problems by
> using polynomial algebra, that is PolynomialReduce etc, with specified
> variable ordering.
>
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.akikoz.net/~andrzej/
> http://www.mimuw.edu.pl/~akoz/
>

This is exactly the case.

When FullSimplify is given equational assumptions it
computes a single Groebner basis of the assumptions
(with the DegreeReverseLexicographic monomial order
and with the variables ordered by Sort).
PolynomialReduce with respect to this Groebner basis
is then applied as one of the transformation functions
used to simplify subexpressions.

Best Regards,