TITLE: Asymptotic expansion of the inverse of $x\mapsto x+x^{\small\sqrt2}+x^2$ near zero
QUESTION [12 upvotes]: This is a follow-up to my previous question "Asymptotic expansion of the inverse of $x\mapsto x+x^\phi$ near zero".
Consider a continuous real-valued monotone increasing function $f:\mathbb R^+\to\mathbb R^+$ satisfying $f\big(x+x^{\small\sqrt2}+x^2\big)=x.$

I am interested in an asymptotic expansion of $f(z)$ for $z\to0^+$ in terms of powers of $z$. I was able to find a few initial terms by manually balancing coefficients:
$$f(z)=z-z^{\small\sqrt2}+\sqrt2\;z^{\small\unicode{x202f}2\unicode{x202f}\sqrt2-1}-z^2+\mathcal O\big(z^{\small\unicode{x202f}3\unicode{x202f}\sqrt2-2}\big), \quad z\to0^+.\tag{$\diamond$}$$
Computing next terms in an ad hoc fashion quickly becomes tedious, so I am looking for a more systematic approach that would allow to obtain a general formula for the terms of this series. I expect it to be a mix of integer powers of $z$ and irrational powers involving $\sqrt2$. Also, I would like to know the radius of convergence of that series.
More generally, I am looking for a uniform approach for inverting generalized polynomials of a single variable that may contain both rational and irrational powers.

REPLY [6 votes]: Some thoughts
Conjecture: When $y>0$ is sufficiently small, the unique positive real solution of $x + x^{\sqrt{2}} + x^2 = y$ is given by
\begin{align}
x &= y + \sum_{k=1}^\infty \frac{(-1)^k}{k!} \frac{\mathrm{d}^{k-1}}{\mathrm{d} y^{k-1}}\Big[(y^{\sqrt{2}} + y^2)^k\Big]\\
&= y - (y^{\sqrt{2}} + y^2) + (\sqrt{2}y^{2\sqrt{2}-1} + (2+\sqrt{2})y^{\sqrt{2}+1} + 2y^3) - \cdots \tag{1}
\end{align}
Motivation: The equation is written as $x = y + (-1)(x^{\sqrt{2}} + x^2)$. By the Lagrange reversion theorem (https://en.wikipedia.org/wiki/Lagrange_reversion_theorem),
we guess (1).
Numerical experiment using Maple: When $y = \frac{1}{100}$, the solution of $x + x^{\sqrt{2}} + x^2 = y$ is
$x_0 \approx 0.008704284007$,
while $y + \sum_{k=1}^{20} \frac{(-1)^k}{k!} \frac{\mathrm{d}^{k-1}}{\mathrm{d} y^{k-1}}\Big[(y^{\sqrt{2}} + y^2)^k\Big]
\approx 0.008704284006$.
However, when $y$ is larger, it may fail. For example, $y = 1/2$,
$y + \sum_{k=1}^{5} \frac{(-1)^k}{k!} \frac{\mathrm{d}^{k-1}}{\mathrm{d} y^{k-1}}\Big[(y^{\sqrt{2}} + y^2)^k\Big]\approx -19.46221804$.
So I guess we need sufficiently small $y$.
Further work: 1) What is the ROC of the series (1)? 2) Prove (or disprove) if it is the solution.