
From:  Ralf Hemmecke 
Subject:  Re: FW: data structure vs. mathematical structure (was: [Axiomdeveloper] Graph theory) 
Date:  Tue, 14 Nov 2006 22:39:45 +0100 
Useragent:  Thunderbird 1.5.0.8 (X11/20061025) 
Why does Axiom have distributed and recursive polynomials. Mathematically it's the same thing, so why bother?
This may not be such a good example since I think the difference here is primarily related to how these domains coerce to OutputForm.
I totally disagree :) This is a good example. I forgot where and if this is the example used in the Axiom book but the authors insisted on this point. I do not think this is related only to the coercion to OutputForm.
[snip]
I'm not a specialist of the Polynomial domains/categories in Axiom (which are, I think, the most developed areas in Axiom) so I can not argue about this, but some algorithms depends intimately on the internal representation used.
Think of implementing Buchberger's algorithm for multivariate commutative polynomial rings. In order to test whether one polynomial is reducible with respect to another, one has to access the exponent vector of the leading term. In a distributed format with keeping the terms sorted LT(p) would be a O(1) operation. I cannot think of something of that complexity for recursive polynomials if the term order is chosen to be "degree reverse lexicographical".
Ralf
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