Approximate roots
Patrick Popescu-Pampu
TL;DR
This work develops the theory of p-th approximate roots for monic polynomials in $\mathbf{C}[[X]][Y]$, extending the classical Newton–Puiseux framework to extract detailed local data from plane branches. The central result identifies a distinguished sequence of characteristic approximate roots $f_k=\sqrt[E_k]{f}$ for an irreducible $f$, with precise intersection-number relations $(f,f_k)=\overline B_{k+1}$ and scaled characteristic exponents $\frac{B_i}{E_k}$, tying the algebraic construction to the semigroup $\Gamma(C)$. The development generalizes to semiroots, preserves key local properties, and provides algorithmic and geometric insights, including a pathway to proving the Abhyankar–Moh embedding line theorem via local to global translation. Overall, the paper integrates valuation theory, semigroup generators, and coordinate-invariant invariants to reveal how approximate roots encode the singularity structure and enable embedding results.
Abstract
Given an integral domain $A$, a monic polynomial $P$ of degree $n$ with coefficients in $A$ and a divisor $p$ of $n$, invertible in $A$, there is a unique monic polynomial $Q$ such that the degree of $P-Q^{p}$ is minimal for varying $Q$. This $Q$, whose $p$-th power best approximates $P$, is called the $p$-th approximate root of $P$. If $f \in \mathbf{C}[[X]][Y]$ is irreducible, there is a sequence of characteristic approximate roots of $f$, whose orders are given by the singularity structure of $f$. This sequence gives important information about this singularity structure. We study its properties in this spirit and we show that most of them hold for the more general concept of semiroot. We show then how this local study adapts to give a proof of Abhyankar-Moh's embedding line theorem.