Karim Belabas on Sun, 30 Oct 2022 14:47:46 +0100

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Re: arithmetic operations with t_INFINITY

* Aurel Page [2022-10-30 13:45]:
> On the other hand, I would like to be the devil's advocate and note that in
> standard definitions of valuations, what is important is that the codomain
> is a totally ordered *abelian group* augmented by oo in which the addition
> law is extended by a+oo = oo for all a. For that point of view, it would be
> natural to define addition
> a + (+oo) = +oo for all a in Z union {+oo}. (I don't think we have
> non-integer valued valuations, do we?)
> and maybe
> a + (-oo) = -oo for all a in Z union {-oo}, since poldegree is treated as
> minus a valuation,
> but not +oo + -oo or multiplications.
> This way, the defining property of valuations: v(xy) = v(x)+v(y) could
> actually be checked.
> That said, I would not push strongly to include those operations.

Then you would certainly want to check that v(x^n) = n*v(x) or vice-versa,
so multiplication/division by an integer is required.

This also points to an extension to t_FRAC as well, i.e. we now have to
support Q union {-oo, oo} plus associated operations (including all gs / sg
variants of course).

And now it becomes harder to prevent such objects from ending up as
t_POL coefficients (which will produce errors on most operations but of
course we must check that it doesn't trigger obscure SEGs), etc.

A can of worms...


Karim Belabas, IMB (UMR 5251), Université de Bordeaux
Vice-président en charge du Numérique
T: (+33) 05 40 00 29 77; http://www.math.u-bordeaux.fr/~kbelabas/