$\newcommand{\kk}{\mathbb{K}} \newcommand{\rr}{\mathbb{R}} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\supp}{supp}$Let $f$ be a power series in $n$ variables $x_1, \ldots, x_n$ over a field $\kk$. If $f$ is regular in $x_n$, i.e. $f(0, \ldots, 0, x_n) \not\equiv 0$, then there is a unique Weierstrass polynomial (with respect to $x_n$), say $w$, and an invertible power series $u$ associated to $f$ such… Continue reading Support of Weierstrass Quotients and Remainders
Tag: Power Series
Approximate roots of polynomials – part 1
$\DeclareMathOperator{\charac}{char}$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… Continue reading Approximate roots of polynomials – part 1
Division with power series
\(\DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\In}{In} \DeclareMathOperator{\ld}{Ld} \DeclareMathOperator{\kk}{\mathbb{K}} \DeclareMathOperator{\nd}{ND} \DeclareMathOperator{\rnpos}{\mathbb{R}^n_{> 0}} \DeclareMathOperator{\rnzero}{\mathbb{R}^n_{\geq 0}} \DeclareMathOperator{\rr}{\mathbb{R}} \DeclareMathOperator{\scrB}{\mathcal{B}} \DeclareMathOperator{\scrI}{\mathcal{I}} \DeclareMathOperator{\scrJ}{\mathcal{J}} \DeclareMathOperator{\supp}{Supp} \DeclareMathOperator{\znplusonezero}{\mathbb{Z}^{n+1}_{\geq 0}} \DeclareMathOperator{\znplusonepos}{\mathbb{Z}^{n+1}_{> 0}} \DeclareMathOperator{\znpos}{\mathbb{Z}^n_{> 0}} \DeclareMathOperator{\znzero}{\mathbb{Z}^n_{\geq 0}} \DeclareMathOperator{\zz}{\mathbb{Z}}\)In this post we look at division in the ring \(\hat R := \kk[[x_1, \ldots, x_n]]\) of formal power series over a field \(\kk\). In contrast to division in polynomial rings discussed in… Continue reading Division with power series