$\DeclareMathOperator{\charac}{char}$
I am going to quickly recall the notion of approximate roots of polynomials due to Abhyankar and Moh, as stated in [Russell, 1980], and going to add a few remarks on the case that the corresponding exponent is not invertible.
Let $A$ be a ring (with an identity) and $f \in A[x]$, where $x$ is an indeterminate, be a monic polynomial of degree $n$, and $d$ be a divisor of $n$. In the case that $d$ is invertible in $A$, the $d$-th approximate root of $f$ is defined via the following result.
Theorem 1 (Abhyankar and Moh, 1973). Assume $d$ is invertible in $A$. Then there is a unique monic polynomial $g \in A[x]$ of degree $n/d$ such that $\deg(f – g^d) < n – n/d$.
Proof. Pick any monic polynomial $\phi$ of degree $n/d$. Then there are unique polynomials $\phi_i \in A[x]$, such that $\deg \phi_i < n/d$, and
\[f = \phi^d + \sum_{i = 1}^d \phi_i \phi^{d-i}\]
Then it is straightforward to check that $g := \phi + (1/d)\phi_1$ satisfies all the requirements including the uniqueness.
Question. What happens if $d$ is not invertible in $A$? More specifically, assume $d$ divides $\charac(A)$. Is there a notion of approximate root in this scenario?
So assume $d = d’p^k$, where $p := \charac(A) > 0$, $k > 0$, and $p$ does not divide $d’$.
Case 1: $d’ = 1$.
Say $n = n’p^s$ where $p$ does not divide $n’$, and $d = p^k$, $k <= s$. Let $g = \sum_i b_i x^{e – i}$, where $e := n/d = mp^{s-l}$. Then $g^d = \sum_i b_i^d x^{d(e-i)}$, so that the subtraction of $g^d$ from $f$ only impacts coefficients of $x^{n – ip^k}$, $i = 0, \ldots, n’p^{s-k}$. So one possible way to define the $p^k$-th approximate root of $f = x^n + \sum_i a_ix^{n-i}$ as
\[g := x^e + \sum_i b_i x^{e – i}\]
where each $b_i$ is an $p^{k}$-th approximate root of $a_{ip^k}$, provided of course that the $p^k$-th approximate roots make sense for the coefficients of $f$.
Case 2: $d’ > 1$.
In this case, one approach is to first take the (unique) $d’$-th approximate root $g’$ of $f$, and then to take the $p^k$-th approximate root of $g’$ if possible. Another way to take the $p^k$-th approximate root first, and then take the $d’$-th approximate root.
Need to explore: what makes more sense? In a reasonable case they should yield the same answer, or it might help to ensure uniqueness. Also, consider the approximations to the power series roots of polynomials in terms of Hamburger-Noether expansions, as described in [Russell, 1980] – what do they correspond to?
References
- [Abhyankar and Moh, 1973] Shreeram Shankar Abhyankar and Tzuong-Tsieng Moh, Newton-Puiseux expansions and generalized Tschrinhausen transformation I and II, Journal für die reine und angewandte Mathematik, 1973.
- [Russell, 1980] Peter Russell, Hamburger-Noether Expansions and Approximate Roots of Polynomials, Manuscripta mathematica, 1980.