Let us assume we want to investigate a family of functions $$R(x) = \sqrt[k]{Q(x)}$$ Where $Q(x)$ is polynomial. We want to to approximate $R(x)$ with polynomial: $R(x) \approx P(x)$
In other words $P(x)$ ("nearly" in some sense) solves the equation $P(x)^k = Q(x)$ for a set of $x$ and a polynomial $Q(x)$ with the extra requirement that $P(x)$ needs be a polynomial.
What would be useful approaches for this problem?
Own work: I used a matrix representation of polynomials together with matrix logarithms to calculate $${\bf M_P} = \exp\left(\frac{\log({\bf M_R})}{k}\right)$$ And it turns out to also become a matrix representation for a polynomial.
Just to show that it can give sensible results: on the function $R(x) = \sqrt{(1+2x)^2} = \left|1+2x\right|$ :
