Hilbert’s Nullstellensatz: Proofs and Applications
Author: Ryan Walter
Table of Contents:
1. Introduction
2. Vocab
3. Theorem
a. Weak
b. Strong
4. Applications
5. Sources
Introduction:
Hilbert's Nullstellensatz is a relationship between algebra and geometry that was discovered by David Hilbert in 1900. Nullstellensatz is a German word that translates roughly to “Theorem of Zeros” or more precisely, “Zero Locus Theorem.” The Nullstellensatz is a foundational theorem that greatly advanced the study of algebraic geometry by proving a strong connection between geometry and a branch of algebra called commutative algebra. Both the Nullstellensatz and commutative algebra focus heavily on ‘rings,’ which will be defined in the vocabulary section.
Vocab:
A polynomial ring is defined as R[x] = R0x0 + R1x1+…+Rnxn, where R0, R1, … Rn are all coefficients in R. This polynomial ring is not a function and these x’s are not replaced by numbers; they are a symbol rather than a value. . Usually, when and R term is zero, the entire term is omitted.
Example: R(x) = 1 + 2x + 0x2 + 0x3 + 0x4 + 3x5 can be written as R(x) = 1 + 2x + 3x5
An two-sided ideal, or simply ideal, of a ring is a special type of ring where any two numbers that are part of the set R, are also part of the set I when added together (a,b ∈ I, a+b ∈ I) and when a number from ring I and a number from the ring R are multiplied together, the product are a part of ring I. (a ∈ I, r ∈ R, ar ∈ I). In this definition, R is any given ring and I is a subset of R.
The ideal can be split further into left and right ideals, where a left ideal is found when sL∈ L, and a right ideal is found when Rs ∈ R. The order of multiplication is significant because the ideals are often displayed as matrices, and the order of multiplication is significant when multiplying matrices. In order for a ideal to be two-sided, it must be both a right and left ideal. In this definition, R is a set that is right-handed, L is a set that is left-handed, and s is a subset of L and R.
For example, {0} is an ideal for every ring, and is known as the trivial ideal.
The matrix below is the left ideal for every 2x2 matrix with real numbers.
0 1 0 1
Proof: Given the matrix of the set of R, we check by verifying sL∈ L
s L sL 0 1 * a b = 0a+1c 0c+1d = c d 0 1 c d 0a+1c 0c+1d c d
sL only has two elements, c and d, which are elements of L. Therefore, this s is a left ideal for all 2x2 matrices.
The matrix below is the right ideal of a ring for all 2x2 matrix with real numbers.
1 1 0 0
Proof: Given the matrix of the set of R, we check by verifying Lr ∈ L
L r Lr a b * 1 1 = 1a+0b 1a+0b = a a c d 0 0 1c+0d 1c+0d c c
Lr only has two elements, a and c, which are elements of L. Therefore, this s is a right ideal for all 2x2 matrices.
Theorem:
Applications:
Sources:
