Author: pinaki

Support of Weierstrass Quotients and Remainders

$\newcommand{\kk}{\mathbb{K}} \newcommand{\rr}{\mathbb{R}} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\supp}{supp}$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

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

Bézout’s theorem in Bézout’s eyes – part 1

$\newcommand{\kk}{\mathbb{K}}$This is the first of a number, hopefully greater than one, of posts in which I record my understanding of Étienne Bézout’s approach towards his theorem on number of (isolated) solutions of (square) polynomial systems. My source is Eric Feron’s translation (General Theory of Algebraic Equations, Princeton University Press, 2006) of Bézout’s Théorie Générale des… Continue reading Bézout’s theorem in Bézout’s eyes – 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)

BKK bound – an “elementary proof”

Some comments of Jonathan Korman made me realize that it would have been good to include in How Many Zeroes? an “elementary” proof of the Bernstein-Kushnirenko formula via Hilbert polynomials. This approach only yields a “weak” version of the bound since it applies only to the case that the number of solutions is finite. Nevertheless,… Continue reading BKK bound – an “elementary proof”

Bernstein-Kushnirenko and Bézout’s theorems (weak version)

$\DeclareMathOperator{\conv}{conv} \newcommand{\dprime}{^{\prime\prime}} \DeclareMathOperator{\interior}{interior} \newcommand{\kk}{\mathbb{K}} \newcommand{\kstar}{\kk^*} \newcommand{\kstarn}{(\kk^*)^n} \newcommand{\kstarnn}[1]{(\kk^*)^{#1}} \DeclareMathOperator{\mv}{MV} \newcommand{\pp}{\mathbb{P}} \newcommand{\qq}{\mathbb{Q}} \newcommand{\rnonnegs}{\mathbb{r}^s_{\geq 0}} \newcommand{\rr}{\mathbb{R}} \newcommand{\scrA}{\mathcal{A}} \newcommand{\scrL}{\mathcal{L}} \newcommand{\scrP}{\mathcal{P}} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\vol}{vol} \newcommand{\znonneg}{\mathbb{Z}_{\geq 0}} \newcommand{\znonnegs}{\mathbb{Z}^s_{\geq 0}} \newcommand{\zz}{\mathbb{Z}}$In earlier posts we defined the degree of a projective variety (defined over an algebraically closed field $\kk$) and showed that it can be determined from its Hilbert polynomial. In this post we… Continue reading Bernstein-Kushnirenko and Bézout’s theorems (weak version)

Degree of a variety via Hilbert polynomial

\(\newcommand{\dprime}{^{\prime\prime}} \newcommand{\kk}{\mathbb{K}} \newcommand{\pp}{\mathbb{P}} \newcommand{\qq}{\mathbb{Q}} \newcommand{\rr}{\mathbb{R}} \newcommand{\zz}{\mathbb{Z}}\)As presented in the preceding post, the degree of a subvariety \(X\) of a projective space \(\pp^n\) is the number of points of intersection of $X$ and a “generic” $n – m$ dimensional linear subspace of $\pp^n$, where $m:= \dim(X)$. In this post we show that the degree of a… Continue reading Degree of a variety via Hilbert polynomial

Degree of a projective variety

$\DeclareMathOperator{\codim}{codim} \newcommand{\dprime}{^{\prime\prime}}\newcommand{\kk}{\mathbb{K}} \newcommand{\local}[2]{\mathcal{O}_{#2, #1}} \newcommand{\pp}{\mathbb{P}} \newcommand{\qq}{\mathbb{Q}} \newcommand{\rr}{\mathbb{R}} \DeclareMathOperator{\res}{Res} \newcommand{\scrL}{\mathcal{L}} \DeclareMathOperator{\sing}{Sing} \newcommand{\zz}{\mathbb{Z}}$The degree of a subvariety $X$ of a projective space $\pp^n$ defined over an algebraically closed field $\kk$ is the number of points of intersection of $X$ and a “generic” linear subspace of $\pp^n$ of “complementary dimension”, i.e. of dimension equal to $n – \dim(X)$.… Continue reading Degree of a projective variety

Notes (part 2): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis

\(\newcommand{\ff}{\mathbb{F}} \DeclareMathOperator{\ord}{ord} \newcommand{\qq}{\mathbb{Q}} \newcommand{\rr}{\mathbb{R}} \newcommand{\rnneg}{\rr_{\geq 0}} \newcommand{\rpos}{\rr_{> 0}} \newcommand{\zz}{\mathbb{Z}} \newcommand{\znneg}{\zz_{\geq 0}} \)Given a normed (abelian) group \(A,\) the following are some important subsets of \(\prod_{i=1}^\infty A\): It is clear that \(A^{(\infty)} \subseteq c(A) \subseteq b(A).\) If \(A\) is a ring/algebra over a field/module over a ring, each of these sets inherits the structure from \(A.\)… Continue reading Notes (part 2): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis