Revisiting one of my older questions, I've decided to try to tackle a simpler version of the problem, this time without the square root coefficients.
Let $x_0$ be a real number such that it satisfies the equation $$x_0+x_0^{1/2}+x_0^{1/3}+\cdots+x_0^{1/n}=1$$ for a natural number $n$. What is the minimal polynomial in $\mathbb{Z}[x]$?
Of course, this is possible by brute force: isolating the smallest power of $x$ then raising both sides by its reciprocal and repeating, but it becomes extremely tedious to do when $n$ is large. Also, this does not guarantee that the polynomial obtained is minimal.
This works fine for $n=1,2,3$. The minimal polynomials are, respectively, $$x-1,\quad x^2-3x+1,\quad x^5-8x^4+24x^3-21x^2+10x-1$$ and it may be interesting to note that the sign of the coefficients are alternating.
Is there an efficient way of doing this for the general case?
Let $x=y^L$, where $L$ is the least common multiple of $2,3,\dots,n$. Then you have the two equations, $y^L-x=0,y^L+y^{L/2}+y^{L/3}+\cdots+y^{L/n}-1=0$. The resultant of these two polynomials will be a polynomial satisfied by $x$. I suspect, but am not sure I could prove, that it's the minimal polynomial.
The resultant can be computed as a determinant. Unfortunately, it's the determinant of a $2L\times2L$ matrix, but at least it's a very sparse matrix. I have a hunch the problem is inherently tedious and that there's no efficient way to do it.