Consider the function $f: (0,\infty) \rightarrow \mathbb{R}$ $$ f(X) = \frac{X^n}{K^n + X^n} - \gamma X $$ where $n \in \mathbb{R}$ is an arbitrary power. Assume for simplicity that $1 \leq n \leq 2$, and also that $K > 0$, and $\gamma > 0$.
Why is it hard to find the (real) solutions to $f(X) = 0$ for general $n$? For integer $n$, i.e. $n=1, n=2$ we can obtain algebraic solutions, but the expressions are not very insightful, and for non-integer $n$ this does not seem feasible. Any reference to the type of equation this is or what field of mathematics studies solutions to such equations would also be helpful.
Comment
Numerically we can solve for this equation and find between 1 and 3 solutions depending on the precise value of $n$, $K$ and $\gamma$. The first term is known as the Hill function and the overall the equation models gene expression with positive feedback (the first term) and degradation (the second term).
Solving for $X$ : $$\frac{X^n}{K^n + X^n} - \gamma X =0$$ $$X^n-\frac{1}{\gamma}X^{n-1}+K^n=0$$ If $n$ is an integer and $1\leq n\leq 4$ the solution(s) are known on the form of the combination of a finite number of elementary functions. This is possible because some elementary functions such as Powers, Roots, etc. have been defined and are commonly used.
If $n=5$ the solution(s) can be expressed on the form of the combination of a finite number of elementary functions and special functions : Jacobi theta functions or some hypergeometric functions. This is possible because those special functions where defined and can be used.
To answer to the question raised: In the general case ($n$ not integer), this does not seem feasible because no convenient special functions are available for this use, up to now.