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...]

## Local Rings

FoldUnfold Table of Contents Local Rings Local Rings Definition: A Local Ring is a ring $R$ that has a unique maximal ideal. Recall that a proper ideal $I$ of a ring $R$ is a maximal ideal if there exists no other proper ideals of $R$ containing $I$. In general, maximal ideals need not be unique. For example, consider the ring of integers $\mathbb{Z}$. The ideals $2 … [Read more...]

## The Rational Function Field of an Affine Variety

FoldUnfold Table of Contents The Rational Function Field of an Affine Variety The Rational Function Field of an Affine Variety Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ be a nonempty affine variety. The Rational Function Field of $V$ is defined as $K(V) = \left \{ \frac{f}{g} : f, g \in \Gamma(V), g \neq 0 \right \}$. Elements of $K(V)$ are called … [Read more...]

## The Coordinate Ring of an Affine Variety

FoldUnfold Table of Contents The Coordinate Ring of an Affine Variety The Coordinate Ring of an Affine Variety Recall from the Affine Varieties page that an affine variety is simply an irreducible affine algebraic set. When we have a nonempty affine variety $V$ we can consider what is called the coordinate ring of $V$ which we define below. Definition: Let $K$ be a field … [Read more...]

## Affine Varieties

FoldUnfold Table of Contents Affine Varieties Affine Varieties Definition: Let $K$ be a field and let $X \subseteq \mathbb{A}^n(K)$ be an affine algebraic set. Then $X$ is said to be Reducible if there exists affine algebraic sets $X_1$ and $X_2$ where $X_1, X_2 \neq \emptyset$ and $X_1, X_2 \neq X$ and such that $X = X_1 \cup X_2$. An affine algebraic set $X$ is said to be … [Read more...]

## Corollary to the Hilbert Basis Theorem

FoldUnfold Table of Contents Corollary to the Hilbert Basis Theorem Corollary to the Hilbert Basis Theorem Recall from The Hilbert Basis Theorem page that if $R$ is a Noetherian ring then $R[x]$ is a Noetherian ring. We will now state a few very important corollaries to the Hilbert basis theorem with regards to affine algebraic sets. Corollary 1: Let $R$ be a Noetherian … [Read more...]