$\newcommand{\kk}{\mathbb{K}}$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
Tag: Bezout
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)