$\newcommand{\kk}{\mathbb{K}} \newcommand{\rr}{\mathbb{R}} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\supp}{supp}$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: Polynomials
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
Polynomial division over valued fields – Part II (Stronger universal bases)
$\DeclareMathOperator{\gr}{gr} \DeclareMathOperator{\In}{In} \DeclareMathOperator{\Inn}{\overline{In}} \DeclareMathOperator{\Incoeff}{In_coeff} \DeclareMathOperator{\Inexp}{In_exp} \DeclareMathOperator{\ld}{Ld} \newcommand{\qq}{\mathbb{Q}} \newcommand{\kk}{\mathbb{K}} \DeclareMathOperator{\ord}{ord} \newcommand{\preceqeq}{\preceq_{\equiv}} \newcommand{\rnpos}{\mathbb{R}^n_{> 0}} \newcommand{\rnzero}{\mathbb{R}^n_{\geq 0}} \newcommand{\rr}{\mathbb{R}} \newcommand{\scrB}{\mathcal{B}} \newcommand{\scrK}{\mathcal{K}} \newcommand{\scrM}{\mathcal{M}} \DeclareMathOperator{\supp}{Supp} \newcommand{\znplusonezero}{\mathbb{Z}^{n+1}_{\geq 0}} \newcommand{\znplusonepos}{\mathbb{Z}^{n+1}_{> 0}} \newcommand{\znpos}{\mathbb{Z}^n_{> 0}} \newcommand{\znzero}{\mathbb{Z}^n_{\geq 0}} \newcommand{\zz}{\mathbb{Z}}$In this post we continue the discussion of polynomial division over valued fields from where we left at Part I. We keep the numberings of the environments (example,… Continue reading Polynomial division over valued fields – Part II (Stronger universal bases)
Polynomial division over valued fields – Part I (Chan-Maclagan’s Algorithm for Homogeneous Divisors)
$\DeclareMathOperator{\gr}{gr} \DeclareMathOperator{\In}{In} \DeclareMathOperator{\Inn}{\overline{In}} \DeclareMathOperator{\Incoeff}{In_coeff} \DeclareMathOperator{\Inexp}{In_exp} \DeclareMathOperator{\ld}{Ld} \newcommand{\qq}{\mathbb{Q}} \newcommand{\kk}{\mathbb{K}} \DeclareMathOperator{\ord}{ord} \newcommand{\preceqeq}{\preceq_{\equiv}} \newcommand{\rnpos}{\mathbb{R}^n_{> 0}} \newcommand{\rnzero}{\mathbb{R}^n_{\geq 0}} \newcommand{\rr}{\mathbb{R}} \newcommand{\scrB}{\mathcal{B}} \newcommand{\scrK}{\mathcal{K}} \newcommand{\scrM}{\mathcal{M}} \DeclareMathOperator{\supp}{Supp} \newcommand{\znplusonezero}{\mathbb{Z}^{n+1}_{\geq 0}} \newcommand{\znplusonepos}{\mathbb{Z}^{n+1}_{> 0}} \newcommand{\znpos}{\mathbb{Z}^n_{> 0}} \newcommand{\znzero}{\mathbb{Z}^n_{\geq 0}} \newcommand{\zz}{\mathbb{Z}}$Today we discuss division of polynomials over valued fields, i.e. fields equipped with a (real) valuation, and consider orderings on monomials which incorporate that valuation. The goal is… Continue reading Polynomial division over valued fields – Part I (Chan-Maclagan’s Algorithm for Homogeneous Divisors)
Polynomial division via initial terms – Part I (Homogenization)
\(\DeclareMathOperator{\In}{In} \DeclareMathOperator{\ld}{Ld} \DeclareMathOperator{\kk}{\mathbb{K}} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\rnpos}{\mathbb{R}^n_{> 0}} \DeclareMathOperator{\rnzero}{\mathbb{R}^n_{\geq 0}} \DeclareMathOperator{\rr}{\mathbb{R}} \DeclareMathOperator{\scrB}{\mathcal{B}} \DeclareMathOperator{\scrI}{\mathcal{I}} \DeclareMathOperator{\scrJ}{\mathcal{J}} \DeclareMathOperator{\supp}{Supp} \DeclareMathOperator{\znplusonezero}{\mathbb{Z}^{n+1}_{\geq 0}} \DeclareMathOperator{\znplusonepos}{\mathbb{Z}^{n+1}_{> 0}} \DeclareMathOperator{\znpos}{\mathbb{Z}^n_{> 0}} \DeclareMathOperator{\znzero}{\mathbb{Z}^n_{\geq 0}} \DeclareMathOperator{\zz}{\mathbb{Z}}\)This is a continuation of an earlier post on polynomial division where we looked at division of polynomials via cancelling the leading term with respect to a monomial order. Here we look at “initial… Continue reading Polynomial division via initial terms – Part I (Homogenization)
Polynomial division and Universal bases
Polynomial division \(\DeclareMathOperator{\In}{In} \DeclareMathOperator{\ld}{Ld} \DeclareMathOperator{\kk}{\mathbb{K}} \DeclareMathOperator{\rnpos}{\mathbb{R}^n_{> 0}} \DeclareMathOperator{\rnzero}{\mathbb{R}^n_{\geq 0}} \DeclareMathOperator{\rr}{\mathbb{R}} \DeclareMathOperator{\scrB}{\mathcal{B}} \DeclareMathOperator{\scrI}{\mathcal{I}} \DeclareMathOperator{\scrJ}{\mathcal{J}} \DeclareMathOperator{\supp}{Supp} \DeclareMathOperator{\znpos}{\mathbb{Z}^n_{> 0}} \DeclareMathOperator{\znzero}{\mathbb{Z}^n_{\geq 0}} \DeclareMathOperator{\zz}{\mathbb{Z}}\)In this post we talk about division with respect to polynomials in more than one variables, which is a pretty cute algorithm that changed the face of a big part of mathematics, via e.g. Gröbner bases which… Continue reading Polynomial division and Universal bases
Lüroth’s theorem (a “constructive” proof)
Lüroth’s theorem (Lüroth 1876 for \(k = \mathbb{C}\), Steinitz 1910 in general). If \(k \subseteq K\) are fields such that \(k \subseteq K \subseteq k(x)\), where \(x\) is an indeterminate over \(k\), then \(K = k(g)\) for some rational function \(g\) of \(x\) over \(k\). I am going to present a “constructive” proof of Lüroth’s… Continue reading Lüroth’s theorem (a “constructive” proof)