complex coefficients

[berceanu]

Does PHCpy support solving systems of nonlinear equations with complex variables, or do I have to split them into real and imaginary parts before feeding them to the solver?

[berceanu]

I should perhaps also mentions that the variables sometimes appear as complex conjugate, ie. z1 * z2 *conj(z1) + 2 * |z2|^2 * z1 + ... = 0 .. more equations ..

[janverschelde]

Instead of separating real and imaginary parts, you could consider the polar representation of a complex number x as x = absxexp(iargx) = rxax. The complex conjugate of x is then written as absxexp(i(-argx) = rx(ax^(-1)). The advantage of this representation over the splitting in real and imaginary parts is that the polynomials will remain sparse. The software supports Laurent polynomials, which are polynomials with negative exponents.

[berceanu]

So if I understand correctly, you suggest to replace $x$ by $r a$ and $x^{\star}$ by $r a^{-1}$, where $r = |x|$ and $a = \exp(i \arg(x))$ and solve for $r$ and $a$ as independent variables? Wouldn't that result in the same number of equations for twice as many variables? Also, is there a way to impose the conditions that $r \geq 0$ and $|a|^2 = 1$?

[berceanu]

I think I don't understand the rxax notation, actually.