$\newcommand{\kk}{\mathbb{K}}$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
Category: Degree
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