Adjoint of polynomials and complex numbers

[ahumenberger]

It seems that the implementation of adjoint is neglecting the fact that the unknowns of polynomials can be replaced by complex numbers.

MultivariatePolynomials.jl/src/operators.jl

Line 116 in fe0d7ca

LinearAlgebra.adjoint(t::AbstractTerm) = LinearAlgebra.adjoint(coefficient(t)) * monomial(t)

using DynamicPolynomials
@polyvar x
p = (2-im)*x
adj_p = adjoint(p)
adj_p(x => 2+im)

The above piece of code yields 3+4im but the result should actually be 5.

[saschatimme]

Conjugation is not an algebraic operation, so there is a priori no conjugation of variables.
For me the adjoint is a map between polynomial rings, acting only on the coefficients of the polynomial.
Therefore, I think the result is correct here.

[ahumenberger]

Okay, I see your point. So the difference lies in viewing the polynomial as an element in the polynomial ring or as a function?

[chriscoey]

Coincidentally, I experienced this point of confusion yesterday when taking to Sascha. I think of them as functions too and I was hoping to add complex polynomial extensions to multivariatepolynomials and sumofsquares to enable modeling and solving complex sum-of-squares optimization problems (which involve real-valued/Hermitian complex polynomial functions). I will try to discuss with @blegat and @saschatimme in the next couple days.

How do you intend to use conjugation?

[ahumenberger]

I'm not using it directly, I came across it while facing JuliaAlgebra/DynamicPolynomials.jl#49.
In the end, it does not matter for me as I'm not working with complex numbers but rationals. I was just curios/confused about the implementation of adjoint.

[saschatimme]

Okay, I see your point. So the difference lies in viewing the polynomial as an element in the polynomial ring or as a function?

Yes exactly :)