Re: arithmetic operations with t_INFINITY

[Aurel Page <aurel.page@normalesup.org>]

On 29/10/2022 23:09, Karim Belabas wrote:
> * Max Alekseyev [2022-10-29 22:38]:
> > For this purpose, the function isinfinite(v) (or alike) would be more
> > suitable, or checking the equality type(v) == "t_INFINITY".
> > Testing "v == oo" is not mathematically rigorous [...]
> If is rigorous, because we *define* valuation(0, p) := +oo or
> poldegree(0) := -oo, i.e. those two sentinel objects +oo and -oo are
> returned by specific functions to indicate well-specified conditions.
> (As explained before, those are not "arithmetic objects" that could for
> instance be returned or incorporated into a floating point computation
> as we could expect from a standard such as IEEE 754.)
>
> Cheers,
>
>      K.B.
>
> P.S. oo can also be used as loop or integral endpoint as in
>    intnum(x = 1, oo, 1/x^2)
> or
>    for (k = 1, oo, ...)
Hi,

I would add that since oo was introduced to encode the output of 
valuations, what is important is that it belongs to the *ordered set* Z 
union {oo}. So having a well-defined comparison is actually the most 
important thing about it.

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.

Cheers,
Aurel