Bill Allombert on Wed, 23 Aug 2017 00:18:50 +0200

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Re: Using bnfisintnorm to check if a number is of the form P

On Tue, Aug 22, 2017 at 05:41:41PM -0400, Charles Greathouse wrote:
> I have a homogeneous polynomial P(x,y) and a number n. I would like to find
> the integers (a, b) such that P(a, b) = n.
> The basic idea is to represent P in univariate form P(x,1) and
> call bnfisintnorm(bnfinit(P), n). But I don't know how to interpret the
> output of bnfisintnorm, and the help entry doesn't really say anything.
> For example,
> bnfcertify(K = bnfinit(z^3+2,1))
> bnfisintnorm(K, 25)
> gives
> [-z^2 - 2*z + 1, z + 3]
> but it's not clear to me how to determine from this that 3^3 + 2*(-1)^3 ==
> 25.

By the way you are very lucky here, because there are infinitely many
solution to the norm equation (due to units) and it is very difficult to
find those that can be written as a+b*x in general.

So I would say, bnfisintnorm is not a workable solution when deg P >=3.