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MathGroup Archive 2007

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Re: Limit and Root Objects

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72686] Re: Limit and Root Objects
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Mon, 15 Jan 2007 05:23:31 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <NDBBJGNHKLMPLILOIPPOKEDJFFAA.djmp@earthlink.net> <eocs96$6s7$1@smc.vnet.net>

In article <eocs96$6s7$1 at smc.vnet.net>,
 Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:

> Remember the point is that you cannot define such  
> a function that will be continuous over the entire 6 dimensional (3  
> complex dimensions) space of polynomials of degree 3 with complex  
> roots (we actually normally remove the discriminant from the space).  

But can you give a concrete example? I am interested in knowing the 
locations/structure of the discontinuities for a specific simple 
polynomial of degree 3 with complex coefficients.

> You are using a smaller subspace ( you have just one complex  
> parameter) and over a smaller subspace naturaly it is possible to  
> have a continuous root.
> If you look carefully at Adam's argument you will be easily able to  
> see what must go wrong over the entire space of complex cubics.

Actually, no. I think that I get the essential idea but I am still not 
able to see what must go wrong over the entire space of complex cubics. 
A concrete example would help.
 
> This in fact is a good illustration of what a double edged weapon  
> graphics and animations are in studying mathematics: they can just as  
> easily mislead your intuition and lead you to wrong conclusions as to  
> right ones. Which is one reason why I think one should never rely too  
> much on such tools when teaching mathematics. Proofs are proofs and  
> no number of "convincing animations" and "experimental mathematics"  
> can replace them.

But if you can show an animation where the discontinuity is apparent 
then you would convince me! And I would then try harder to better 
understand Adam's argument.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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