FoldUnfold Table of Contents The Multiplicity of an Affine Plane Curve at a Point The Multiplicity of an Affine Plane Curve at a Point Definition: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. Let $F = F_m + F_{m+1} + ... + F_n$ where each $F_i$ is a form of degree $i$. The Multiplicity of $F$ at $\mathbf{p} = (0, 0)$ is defined to be … [Read more...]

## Simple and Multiple Points of Affine Plane Curves

FoldUnfold Table of Contents Simple and Multiple Points of Affine Plane Curves Simple and Multiple Points of Affine Plane Curves Recall from the Affine Plane Curves page that we said that two polynomials $F, G \in K[x, y]$ are said to be equivalent if there exists a nonzero $\lambda \in K$ such that $F = \lambda G$, and we said that an affine plane curve is an equivalence … [Read more...]

## Affine Plane Curves

FoldUnfold Table of Contents Affine Plane Curves Affine Plane Curves Definition: Let $K$ be a field. Two polynomials $F, G \in K[x, y]$ are said to be Equivalent if there exists a nonzero $\lambda \in K$ such that $F = \lambda G$. This forms an equivalence relation on the set of polynomials in $K[x, y]$. An Affine Plane Curve is an equivalence class of such nonconstant … [Read more...]

## Polynomial Forms

FoldUnfold Table of Contents Polynomial Forms Polynomial Forms Definition: Let $R$ be a ring. A polynomial $F \in R[x_1, x_2, ..., x_n]$ is said to be a Form of Degree $d$ if every term in $F$ is of degree $d$. For example, the following polynomials are forms of degree $2$, $3$, and $4$ respectively: (1) \begin{align} \quad F(x, y, z) = x^2 + xy \quad , \quad G(x, y, z) = … [Read more...]

## Discrete Valuation Rings

FoldUnfold Table of Contents Discrete Valuation Rings Discrete Valuation Rings Definition: A Discrete Valuation Ring (DVR) is an integral domain $R$ with the following properties:1) $R$ is a Noetherian ring.2) $R$ is a local ring.3) The unique maximal ideal of $R$ is a principal ideal. Recall that an ideal $I$ of a ring $R$ is a principal ideal if it is generated by a … [Read more...]