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

[Charles Greathouse <charles.greathouse@case.edu>]

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.

Charles Greathouse

Case Western Reserve University