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Re: Re: Root function

• To: mathgroup at smc.vnet.net
• Subject: [mg51111] Re: [mg51033] Re: [mg51009] Root function
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Tue, 5 Oct 2004 04:36:55 -0400 (EDT)
• References: <NDBBJGNHKLMPLILOIPPOMEDOEDAA.djmp@earthlink.net>
• Sender: owner-wri-mathgroup at wolfram.com

David Park wrote:
> Daniel,
> 
> What is the "canonical ordering in the complex plane"?
> 
> Is it a lexicographic ordering by real and imaginary parts? Or by Abs and
> Arg? Or something else?
> 
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
> 
> From: Daniel Lichtblau [mailto:danl at wolfram.com]
To: mathgroup at smc.vnet.net
> 
> The Root form is a concise way of expressing algebraic numbers via the
> minimal polynomial they satisfy, along with a canonical ordering in the
> complex plane (hence those numbers 1-3 in the second field).
> 
> Daniel Lichtblau
> Wolfram Research

Well, "canonical" may carry the wrong connotations of mathematical 
inevitibility. What I should have stated is that Mathematica uses an 
ordering that canonicalizes; but there is noting intrinsic to the 
mathematics that forces that particular ordering to be used; it is 
simply convenient.

The particulars, as best I recall, are as follows.

(i) Order real roots first, from most negative upward.
(ii) Order complex roots by their real parts (negative to positive).
(iii) for ties e.g. complex cnnjugates, suborder next by magnitude of 
imaginary part, negative before positive. This has the beneficial effect 
of keeping conjugate pairs together.

This may not be entirely correct but should give the idea.

Daniel Lichtblau
Wolfram Research