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Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106054] Re: [mg106011] Re: [mg105989] algebraic numbers
  • From: Robert Coquereaux <robert.coquereaux at gmail.com>
  • Date: Thu, 31 Dec 2009 03:17:22 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <200912300914.EAA17185@smc.vnet.net> <4B3B89A4.2010700@wolfram.com>
  • Reply-to: Daniel Lichtblau <danl at wolfram.com>

"Impossible....Not at all"
I think that one should be more precise:
Assume that x algebraic, and suppose you know (only) its first 50 
digits. Then consider y = x +  Pi/10^100.
Clearly x and y have the same first 50 digits , though y is not 
algebraic.
Therefore you cannot recognize y as algebraic from its first 50 digits !
The quoted comment was in relation with the question first asked by 
hautot.
Now, it is clear that, while looking for a solution x of  some 
equation (or definite integral or...), one can use the answer obtained 
by applying  RootApproximant (or another function based on similar 
algorithms) to numerical approximations of x, and then show that the 
suggested algebraic number indeed solves exactly the initial problem. 
If so, you will indeed have recognized the number x as algebraic, from 
its first N figures.
But this does not seem to be the question first asked by hautot.
Also, if one is able to obtain information, for any N, on the first N 
digits of a real number x, this is a different story... and a 
different question.

Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit :

>
>> To recognize a number x as algebraic, from its N first figures, is 
>> impossible.
>
> Not at all. There are polynomial factorization algorithms based on 
> this notion (maybe you knew that).
>
> Daniel Lichtblau
> Wolfram Research



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