Michel Talon wrote:
> ...snip...
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> What would be a legitimate generalization is an algebraic relation of the
> form P(q,x)=0 which in fact describes a curve of genus 1 while it is hidden
> in this explicit form. Then one needs to resort to algorithms (which exist,
> i know a package exists for Maple) to change variables to good
> variables q and w such that q^2=P(x). But i think this more complicated
> case, which can occur in theory, is less likely to occur in practice than
> the straightforward integration of a square root of a polynomial of degree
> 4 multiplied by a rational function.
>
A common observation in writing a program (like integration which may
reduce to elliptic integrals) is that the usual cases that are described
in texts and references, might take 10% of the code that is needed. The
90% of the code, and much of the thought needed is dealing with the
other cases. I don't know what that might be here, but singularities,
repeated roots, symbols where you expect numbers (e.g. sqrt(x^n +a^n) ).
For example, calculus texts describe integration of rational functions
using partial fractions. They factor the denominator into linear and
quadratic factors. All the examples factor in this way. But a program
must consider the case where the denominator does not factor, and this
is perhaps 90% of the effort.
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