Complex{BigFloat} as a Ring structure

[fabiencorinaldesi1]

Hello,

I am trying to define polynomials where coefficients are complex floating-point numbers.
The problem is that Complex{BigFloat} don't have the Ring or Field structure when using AbstractAlgebra.
Would it be difficult to add the Ring structure to Complex{BigFloat} ?

Thank you in advance for your response.

Fabien

[fingolfin]

In theory this could be added, in practice I am not sure how useful it would be.

What would you like to do with the resulting polynomials? Depending on your use case, either a different coefficient ring (e.g. from Nemo), or a different package providing polynomials (e.g. Polynomials.jl) may be more appropriate.

[fabiencorinaldesi1]

Dear Max,

Thanks for your prompt answer!

I'm using Nemo/AbstractAlgebra to design validated symbolic-numeric algorithms, based on a posteriori validation.

Hence, I first compute purely numerical solutions of the problem (for example, Puiseux series solutions to P(x,y(x))=0),
and then use the validated numerics part (using Nemo/Arb) to certify the solution.

This is why I would prefer to do all the computations in the same library rather than duplicating the algorithms
in 2 different libraries (e.g., Nemo and Polynomials.jl) just because of different data types (floating-point and intervals/balls).

Also, it turns out than the Nemo wrapper of Arb only imports arb/acb types (i.e., balls), and not the underlying
floating-point types (called arf/acf in the C code of Arb).

This is why I was planning to use the "native" Julia types BigFloat and Complex{BigFloat}.
Unfortunately, AbstractAlgebra provides a field structure to the former but not the latter,
and I was wondering if this was planned in the near future.

Many thanks for your help!

Fabien