# How many functions does it take to solve polynomial equations?

It is well known that polynomial equations of degree $$\ge 5$$ cannot be solved by just using addition, multiplication, division and radicals (Abel–Ruffini theorem). So what, who cares? Let's just add extra functions until we can solve them! For example, it turns out that polynomial equations of degree 5 can be solved by adding the Bring Radical to our arsenal.

Let $$k_n$$ be the minimum number of univariate, continuous functions that need to be added to +,-,*,/ in order to solve polynomial equations of degree $$n$$.

We know that $$k_0=0$$, $$k_1 = 0$$, $$k_2=1$$, $$k_3=2$$, $$k_4=2$$, $$k_5=4$$ (not counting $$\sqrt[4]{\cdot} = \sqrt{\sqrt{\cdot}}$$)

• Is there anything of substance known about the sequence $$k_n$$? Asymptotics?

I suppose that due to Sprecher's variant of the Kolmogorov-Arnold Representation Theorem we have an upper bound $$k_n\le 2n$$ (given a monic polynomials of degree $$n$$ with coefficients $$a_k$$ define $$\lambda_m(a_0,\ldots,a_{n-1}) =$$"$$m$$"-th root and apply the theorem)

Secondly, can we expect that there exists a minimal family of functions which is nested? Call $$\mathcal F = \{F_n\mid n\in \mathbb N\}$$ with $$F_n\subset C(\mathbb R, \mathbb R)$$ a minimal family iff (1) $$\forall n: |F_n| = k_n$$ (2) $$F_n$$ allows us to solve all polynomial equations of degree $$n$$.

• Does there exist a minimal family such that $$F_0\subseteq F_1\subseteq F_2 \subseteq F_3 \subseteq \ldots$$?