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Re: Re: "Assuming"

  • To: mathgroup at smc.vnet.net
  • Subject: [mg85818] Re: [mg85792] Re: "Assuming"
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 23 Feb 2008 04:27:37 -0500 (EST)
  • References: <20080221171506.200$2n_-_@newsreader.com> <200802221221.HAA08545@smc.vnet.net>

Andrzej Kozlowski wrote:
> On 21 Feb 2008, at 23:15, David W. Cantrell wrote:
> 
> 
>>[Message also posted to: comp.soft-sys.math.mathematica]
>>
>>Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>>[...]

I intersperse two comments, based on two posts to this thread.

[From Andrzej Kozlowski:]
> Note also one more thing.  Suppose we consider a general rational  
> function p[x]/q[x], (or even a rational function in several  
> variables). Should Mathematica be then required always to try to find  
> if p[x] and q[x] do not have common roots in some algebraic (or even  
> transcendental inthe case of several variables) extension of the  
> rationals? This is far from a trivial problem? [...]
 > Andrzej Kozlowski

The only removable singularities, in this context, are in fact from 
exact divisors. So polynomial gcd extraction (used in, say, Together[]) 
suffices to remove them.

Other cases where numerator and denominator simultaneously vanish will 
give points of indeterminacy. These are not removable.

I'm not sure in what way transcendentals snuck in here. If the 
polynomials have common roots, they are describable as an algebraic set. 
In particular, isolated common roots will be algebraic numbers.


[From David Cantrell:]

>>[...] But I do know of a case where
>>Mathematica goes even further, removing a singularity at which the  
>>function
>>is defined as a number:
>>
>>In[17]:= FullSimplify[UnitStep[-x^2]]
>>Out[17]= 0
>>
>>despite the fact that correctly
>>
>>In[18]:= UnitStep[-x^2] /. x -> 0
>>Out[18]= 1
>>
>>Perhaps the simplification above is considered a bug, perhaps not.
>>
>>David

A feature, really. That is, it's wrong, but FullSimplify can make 
mistakes on measure zero sets. We do not generally regard this 
phenomenon as a bug, though we reconsider on case by case basis.


Daniel Lichtblau
Wolfram Research


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