Introduction

• In these notes, unless otherwise stated, all sets are considered to be proper sets as in ZFC.
• Basic (at least naive) set theory and simple combinatorics knowledge is assumed. Other than that, notes should be self-sufficient.

Note that these notes are still under construction.

Resources Used

• Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox, John Little, and Donal O’Shea

Notation

Same as in the Group Theory and Ring Theory notes with the addition of:

• Instead of x_1, ..., x_n, we may write \bar{x} if the context is clear or the size is not a matter of importance for brevity.

Def. Monomial

A monomial in x_1, ..., x_n is a product of the form

where all of the exponents \alpha_1,...,a_\alpha are non-negative integers. The total degree of this monmial is the sum

Notation. Monomial

Let \alpha = (\alpha_1, ..., \alpha_n) be an n-tuple of non-negative integers, then we define

and for \alpha = (0, ..., 0) we have x^\alpha := 1. Moreover, we also define

which denotes the total degree of x^a.

Def. Polynomial

A polynomial f in x_1, ..., x_n with coefficients in the field \mathbb{F} is a finite linear combination of monomials, denoted:

i.e. f is the sum over a finite number of n-tuples \alpha = (\alpha_1, ..., \alpha_n).

Here k_\alpha does not have any special meaning other than the fact that it just used to index the correspoding coefficient for the n-tuple a.

The set of all all polynomials in x_1, ..., x_n with coefficients in \mathbb{F} is denoted with \mathbb{F}[x_1, ..., x_n] or simply by \mathbb{F}[\bar{x}] if the context is clear.

We usually use the lowercase letters f, g, h, p, q, r to denote polynomials.

The k_\alpha is called the coefficient of the monomial x^\alpha. Unless k_\alpha \neq 0, we say k_a x^a is called a term of the polynomial f. The total degree of f denoted by d(f) is the maximum |\alpha| such that the corresponding k_\alpha \neq 0.

So far, all the definitions we have made should be intuitive.

Thm. Polynomials form a Commutative Ring

The set of polynomials \mathbb{F}[x_1, ..., x_n] for any given field \mathbb{F} forms a commutative ring under the standard polynomial addition and multiplication.

Exercise

Def. Algebraically Closed

Given a field \mathbb{F} if every non-constant polynomial in \mathbb{F}[x] has a root in \mathbb{F} we say \mathbb{F} is algebraically closed.

Notice that, \mathbb{F}[x] is the set of polynomials with single indeterminate x unlike \mathbb{F}[x_1, ..., x_n].

Thm. Fundamental Theorem of Algebra

The field of complex numbers \mathbb{C} is algebraically closed. Therefore, every non-constant polynomial f \in \mathbb{C}[x] has a root in \mathbb{C}.

We will not prove this common theorem here. You might want to check out other resources online for a wide range of proofs. However, we will prove a more general theorem called Hilbert Nullstellensatz later on.

Note that \R is not algebraically closed as x^2 + 1 has no roots in \R

Def. Affine Space

Given a field \mathbb{F} and a positive integer n, we define the n-dimensional affine space over \mathbb{F} to be the set

In particular, we call \mathbb{F}^n the affine line for the n=1 and affine plane for n=2.

Notice that given (a_1, ..., a_n) \in \mathbb{F}^n the polynomial f \in \mathbb{F}[x_1, ..., x_n] gives a function f: \mathbb{F}^n \to \mathbb{F} if we replace each x_i by a_i. This is the fundamental idea behind our connection between the algebra and the geometry.

Thm. Infinite Field Polynomials and Zeroes

Let \mathbb{F} be an infinite field, and let f \in \mathbb{F}[x_1, ..., x_n]. Then f = 0 (in \mathbb{F}[x_1, ..., x_n]) if and only if f: \mathbb{F}^n \to \mathbb{F} is the zero function.

Corollary. Polynomial Equality in Infinite Fields

Let \mathbb{F} be an infinite field and let f,g \in \mathbb{F}[x_1, ..., x_n]. Then f=g (in \mathbb{F}[x_1, ..., x_n]) if and only if f and g are the same function.

Exercise

Exercises

Def. (Affine) Variety

Let \mathbb{F} be a field and f_1, ..., f_s \in \mathbb{F}[x_1, ..., x_n], then we define the (affine) variety of f_1, ..., f_s as:

so that \mathbf{V}(f_1, ..., f_s) \subseteq \mathbb{F}^n is the set of all solutions to:

Notice how we used \bar{a} to denote (a_1, ..., a_n) and f(\bar{a}) to denote f(a_1, ..., a_n) where each a_i \in \mathbb{F}.

IMPORTANT: From now on, unless otherwise stated, by variety we will mean the affine variety defined here.

We will commonly use the letter V, W to denote the varieties.

Notice that the graphs of polynomials functions y = f(x) is the variety \mathbf{V}(y - f(x)).

Thm. Basic Properties of Varieties

Let V, W \subseteq \mathbb{F}^n be varieties, then so are V \cup W and V \cap W. Moreover, if V = \bold{V}(f_1,...,f_s) and W = \bold{V}(g_1, ..., g_t), then

1. V \cap W = \bold{V}(f_1,...,f_s,g_1,...,g_t), and
2. V \cup W = \bold{V}(f_i g_j : i = 1,...,s \> \land \> j = 1,...,t).

Exercises

Def. Ideal

A subset S \subseteq \mathbb{F}[\bar{x}] is an (polynomial) ideal of \mathbb{F}[\bar{x}] if

1. 0 \in I
2. f,g \in I \implies f + g \in I
3. f \in I \enspace \land \enspace h \in \mathbb{F}[\bar{x}] \implies hf \in I.

Def. Ideal Generators

Let f_1, f_2, ..., f_s \in \mathbb{F}[\bar{x}], then we define the ideal generated by f_1, ..., f_s as

Exercise that \Braket{f_1, ..., f_s} is indeed an ideal.

Moreover, we say an ideal I is finitely generated if there exists f_1, ..., f_s such that I = \Braket{f_1, ..., f_s} and we say f_1, ..., f_s are a basis of I.

Notice how ideal generated by polynomials is kind of similar to the span of finite number of vectors. Especially with the further facts we will talk about talker such as every ideal in \mathbb{F}[\bar{x}] is finetely generated with a cannoncial basis called Gröbner basis.

Thm. Basis and Variety

For two basis for an ideal in \mathbb{F}[\bar{x}] such that \Braket{f_1, ..., f_s} = \Braket{g_1, ..., g_t}, we have

Exercise

Def. Ideal Variety

Let V \subseteq \mathbb{F}^n be a (affine) variety, then we define the ideal (of) variety V as

Exercise the fact that \bold{I}(V) is indeed an ideal.

Thm. Ideals and Varieties

Let f_1, ..., f_s \in \mathbb{F}[x_1, ..., x_n], then

...and the equality need not to occur (exercise).

Thm. Ideals and Varieties

Let V, W be varieties in \mathbb{F}^n, then

1. V \subseteq W if and only if \bold{I}(V) \supseteq \bold{I}(W).
2. V = W if and only if \bold{I}(V) = \bold{I}(W).

Before we proceed, here are some quetions to ponder about:

• (Ideal Membership) For a finitely generated ideal I, is there any algorithm which decides if any f \in I.
• (Nullstellensatz) What is the exact relation between \Braket{f_1, ..., f_s} and \bold{I}(\bold{V}(f_1, ..., f_s)).

Def. Leading Term

Let 0 \neq f \in \mathbb{F}[x] such that

Notice how we indexed the coefficients

where a_i \in \mathbb{F} and a_0 \neq 0. We say a_0 x^m is the leading term of f denoted with

Notice that if f and g are nonzero polynomials we have

d(f) \leq d(g) \iff \text{LT}(f) \mid \text{LT}(g).

Thm. Division Algorithm

Let f,g \in \mathbb{F}[x] where g \neq 0, then there exists unique q,r \in \mathbb{F}[x] such that

where either r = 0 or d(r) < d(g). The following program realize this:

<- f, g
 
q <- 0
r <- f
 
WHILE r != 0 AND (LT(g) | LT(r))
    q <- q + LT(r) / LT(g)
    r <- r - (LT(r) / LT(g))g
ENDWHILE
 
-> q, r

Thm. Maximum Number of Roots

Let 0 \neq f \in \mathbb{F}[x], then f has at most d(f) roots in \mathbb{F}.

Exercise, use induction and division fact.

Thm. Ideals of \mathbb{F}[x]

Every ideal of \mathbb{F}[x] can be written as \Braket{f} for some f \in \mathbb{F}[x]. Therefore, every ideal of \mathbb{F}[x] is principal. Moreover, such f is unique up to multiplication by a non-zero constant in \mathbb{F}.

Exercise, use the division fact for a non-zero ideal.

Def. \text{GCD}(\cdot, \cdot)

Let f,g \in \mathbb{F}[x], then we say h \in \mathbb{F}[x] is their greatest common divisor if

1. h \mid f,g
2. p \mid f,g \implies p \mid h

Such h is denoted with

Thm. Basic Properties of \text{GCD}(\cdot, \cdot)

Let f,g \in \mathbb{F}[x], then

1. \text{GCD}(f,g) always exists and is unique up to multiplication by a non-zero constant in \mathbb{F}.
2. \text{GCD}(f,g) is a generator of the ideal \Braket{f,g}.
3. The following algorithm generates h = \text{GCD}(f,g)

Exercise, for (1) and (2) use the corresponding principal ideal.

<- f, g
 
h <- f
s <- g
 
WHILE s != 0
    rem <- remainder of h/s
    h   <- s
    s   <- rem
ENDWHILE
 
-> h

Notice that the algorithm uses the division fact f = qg + r so that

since \Braket{f-qg, g} = \Braket{f,g}.

So, to find a GCD, we repeat the division algorithm until the remainder is 0.

Thm. \text{GCD}(\cdot, \cdot, \cdots)

Let f_1, ..., f_s \in \mathbb{F}[x], then a greatest common divisor of f_1, ..., f_s is defined as h \in \mathbb{F}[x] such that

1. h \mid f_1, ..., f_s
2. p \mid f_1, ..., f_s \implies p \mid h

denoted \text{GCD}(f_1, ..., f_s).

Thm. Basic Properties of Generalized \text{GCD}

Let f_1, ..., f_s \in \mathbb{F}[x], then

1. \text{GCD}(f_1, ..., f_s) exists and unique up to multiplication by a non-zero constant in \mathbb{F}.
2. \text{GCD}(f_1, ..., f_s) is a generator of the ideal \Braket{f_1, ..., f_s}.
3. For s \geq 3, we have \text{GCD}(f_1, ..., f_s) = \text{GCD}(f_1, \text{GCD}(f_2, ..., f_s)).

Exercise. The proof is similar to \text{GCD}(\cdot, \cdot).

Def. Monomial Order

A monomial order > on \mathbb{F}[x_1, ..., x_n] is any relation > on \N^n, or equivalently, any relation on the set of monomials x^\alpha that satisfies, for \alpha, \beta \in \N^n

1. > is a total (or linear) order on \N^n
2. \alpha > \beta implies \alpha + \gamma > \beta + \gamma for all \gamma \in \N^n
3. > is a well ordering on \N^n i.e. every non-empty subset of \N^n has a smallest element under >.

Due to Dickson's Lemma (which we will prove later) we could replace (3) with \alpha \geq 0 as (1) and (2) implies (3).

Def. Lexicographic Order

Let \alpha = (\alpha_1, ..., \alpha_n) and \beta = (\beta_1, ..., \beta_n) where \alpha,\beta \in \N^n. The lexiocographic (or lex) order \succ is defined as \alpha \succ \beta if and only if the leftmost non-zero entry is positive in \alpha - \beta .

For example (1,2, 0) \succ (0,3,4) since (1,2,0) - (0,3,4) = (1,-1,4).

From now on, we will write x^\alpha \succ x^\beta if \alpha \succ \beta. We will also use \succcurlyeq to denote \alpha \succ \beta or \alpha = \beta.

Thm. Lexicographic is Monomial

The lexicographic order \succ is a monomial ordering.

Def. Graded Lex Order

Let \alpha = (\alpha_1, ..., \alpha_n) and \beta = (\beta_1, ..., \beta_n) where \alpha,\beta \in \N^n. The graded lexicographic (grlex) order \gg is defined as \alpha \gg \beta if and only if |\alpha| \geq |\beta|, or |\alpha|=|\beta| and \alpha \succ \beta.

Def. Graded Reverse Lex Order

Let \alpha = (\alpha_1, ..., \alpha_n) and \beta = (\beta_1, ..., \beta_n) where \alpha,\beta \in \N^n. The graded reverse lexicographic (grevlex) order \gg is defined as \alpha \gg \beta if and only if |\alpha| \geq |\beta|, or |\alpha|=|\beta| and the rightmost non-zero entry of \alpha - \beta is negative.

Def. Fundamental Definitions

Let f = \sum_\alpha c_\alpha x^\alpha be a polynomial in \mathbb{F}[x_1, ..., x_n] and let > be a monomial order, then we define

1. multidegree of f as: md(f) := \max(\alpha : c_\alpha \neq 0)
2. leading coefficient of f as: \text{LC}(f) := c_{md(f)}
3. leading monomial of f as: \text{LM}(f) := x^{md(f)}
4. leading term of f as: \text{LT}(f) := \text{LC}(f) \cdot \text{LM}(f)

Notice that \max is w.r.t. the monomial order >.

Def. Basic Properties of Monomials

Let f, g \in \mathbb{F}[x_1, ..., x_n] be non-zero polynomials and, then

1. md(fg) = md(f) + md(g)
2. f + g \neq 0 \implies md(f+g) \leq \max(md(f), md(g)). Moreover, they are equal if in addition to f+g\neq 0, we have md(f) \neq md(g).

where < and \leq are the monomial orders. From now on, assume a fixed monomial order > is selected unless otherwise stated.

Thm. Division Algorithm in \mathbb{F}[x_1,...,x_n]

Let F = (f_1, ..., f_s) be an ordered s-tuple of polynomials in \mathbb{F}[x_1, ..., x_n], then every f \in \mathbb{F}[x_1,...,x_n] can be written as

where a_i, r \in \mathbb{F}[x_1, ..., x_n], and either the remainder

• r = 0, or
• r is a linear combination of monomials with coefficients in \mathbb{F}, none of which is divisable by any of \text{LT}(f_1), ..., \text{LT}(f_s).

Moreover, if a_i f_i \neq 0, then md(f) \geq md(a_i f_i).

The following algorithm realize this:

<- (f_1,...,f_s), f
 
(a_1,...,a_s) <- (0,...,0)
p <- f
 
WHILE p != 0
    i <- 1
    divisionOccured <- false
 
    WHILE i <= s AND divisionOccured = false
        IF LT(f_i) | LT(p)
            a_i <- a_i + LT(p) / LT(f_i)
            p   <- p  - (LT(p) / LT(f_i))f_i
            divisionOccured <- true
        ELSE
            i <- i + 1
        ENDIF
    ENDWHILE
 
    IF divisionOccured = false
        r <- r + LT(p)
        p <- p - LT(p)
    ENDIF
ENDWHILE
 
-> (a_1,...,a_s), r

Def. Monomial Ideal

An ideal I \subseteq \mathbb{F}[x_1, ..., x_n] is a monomial ideal if there exists (possibly infinite) basis of monomials so that

Lemma. Membership via Division

Let I = \Braket{x^\alpha : \alpha \in A} be a monomial ideal, then x^\beta \in I if and only if x^\alpha \mid x^\beta for some \alpha \in A.

Lemma. Membership and Equivalence

Let I be a monomial ideal and f \in \mathbb{F}[x_1,...,x_n], then the following are equivalent

1. f \in I.
2. Every term of f lies in I.
3. f is a \mathbb{F}-linear combination of the monomials in I.

Therefore, two monomial ideals are the same if and only if they contain the same monomials.

Thm. Dickson's Lemma

Let I = \Braket{x^\alpha : \alpha \in A} \subseteq \mathbb{F}[x_1,...,x_n] be a monomial ideal, then

Therefore every monomial ideal has a necessarily finite monomial basis.

Def. \text{LT}(I)

Let \Braket{0} \neq I \subseteq \mathbb{F}[x_1,...,x_n] be an ideal, then we define the set of leading terms of I as

Thm. \Braket{\text{LT}(I)}

Let I \subseteq \mathbb{F}[x_1,...,x_n] be an ideal, then

1. \Braket{\text{LT}(I)} is an monomial ideal, and
2. there exists g_1, ..., g_s \in I such that \Braket{\text{LT}(I)} = \Braket{\text{LT}(g_1), ..., \text{LT}(g_s)}.

Thm. Hilbert Basis Theorem

Every ideal I \subseteq \mathbb{F}[x_1,...,x_n] has a finite basis, so that

for some g_1, ..., g_s \in I.

Def. Gröbner Basis

Fix a monomial order and let I be an ideal, then a finite subset G = \Set{g_1, ..., g_s} \subseteq I is said to be a Gröbner basis (Gr-basis) or standard basis if

Equivalently, \Set{g_1, ..., g_s} is a Gr-basis of I if and only if the leading term of any element of I is divisible by one of \text{LT}(g_i).

Thm. Existence of Gr-basis

Every non-zero ideal I \subseteq \mathbb{F}[x_1,...,x_n] has a Gröbner basis. Moreover, any Gröbner basis of I is a basis of I.

Thm. ACC

Let I_1 \subseteq I_2 \subseteq \cdots be an asceding chain of ideals in \mathbb{F}[x_1, ..., x_n], then there exists an N \geq 1 so that I_N = I_{N+i} for all i \in \N.

Def. \bold{V}(I)

Let I \subseteq \mathbb{F}[x_1, ..., x_n] be an ideal, then we define the variety of I as

Thm. Varieties are Determined by Ideals

Let I be an ideal, then \bold{V}(I) is indeed an (affine) variety. Moreover, if I = \Braket{f_1, ..., f_s}, then

Therefore, varieties are determined by ideals.

Thm. Normal Form

Let G = \Set{g_1, ..., g_s} be a Gr-basis for an ideal I \subseteq \mathbb{F}[x_1, ..., x_n] and f \in \mathbb{F}[x_1,...,x_n], then there exists an unique r \in \mathbb{F}[x_1,...,x_n] such that

1. no term of r is divisible by any of \text{LT}(g_1), ..., \text{LT}(g_s), and
2. there exists g \in I so that f = g + r

In particular, r is the remainder on division of f by G (no matter how the elements of G are listed) using the division algorithm.

Such r is called the normal form of f.

Corollary. Membership via Division

Let G = \Set{g_1, ..., g_s} be a Gr-basis for an ideal I \subseteq \mathbb{F}[x_1, ..., x_n] and f \in \mathbb{F}[x_1, ..., x_n], then f \in I if and only if G \mid f i.e. division of f by G is zero.

Notation. Remainder

Let F = (f_1, ..., f_s) be an orderer s-tuple. We will denote the remainder of f by division of F with

Notice that if \Set{f_1, ..., f_s} is a Gr-basis for \Braket{f_1, ..., f_s}, the order of F doesn't matter.

Def. \text{LCM} of Monomials

Let f,g \in \mathbb{F}[x_1, ..., x_n] be non-zero polynomials where md(f) = \alpha and md(g) = \beta, and

We call x^\gamma the least common multiplier of \text{LM}(f) and \text{LM}(g), denoted with

Def. \bold{S}-polynomial

Let f,g \in \mathbb{F}[x_1,...,x_n], then we define their \bold{S}-polynomial as

where x^\gamma = \text{LCM}(\small \text{LM}(f), \text{LM}(g) \normalsize).

Note that we are inverting the leading coefficients here as well.

Thm. Buchberger’s Criterion

Let I be an ideal, then a basis G = \Set{g_1, ..., g_s} is a Gr-basis for I if and only if for all pairs i \neq j, we have G \mid \bold{S}(g_i,g_j).

Thm. Buchberger's Algorithm

We know now that every non-zero ideal in \mathbb{F}[x_1, ..., x_n] has a Gröbner basis. In order to construct such basis we have Buchberger's Algorithm.

Let I = \Braket{f_1,...,f_s} \neq \Braket{0} be a polynomial ideal, then a Gröbner basis for I can be constructed in a finite number of steps by the algorithm:

<- F = (f_1, ..., f_s)
 
G <- F
 
DO
    G' <- G
 
    FOR {p, q} where p != q in G'
        S <- remainder of S(p,q) by G'
 
        IF S != 0
            G <- G ∪ {S}
        ENDIF
    ENDFOR
WHILE G = G'
 
-> G

where G is a Gröbner basis for I with F \subseteq G.

Thm. Lemma

Let G be a Gröbner basis for the polynomial ideal I and p \in G such that \text{LT}(p) \in \Braket{\small\text{LT}(G - \{p\})}, then G - \{p\} is also a Gröbner basis for I.

Def. Minimal Gröbner Basis

A minimal Gröbner basis G for a polynomial ideal I is a Gröbner basis for I such that for all p \in G

1. \text{LC}(p) = 1, and
2. \text{LT}(p) \notin \Braket{\small\text{LT}(G - \{p\})}.

Def. Reduced Gröbner Basis

A reduced Gröbner basis G for a polynomial ideal I is a Gröbner basis for I such that for all p \in G

1. \text{LC}(p) = 1, and
2. no monomial of p lies in \Braket{\small\text{LT}(G - \{p\})}.

Thm. Ideal Equality

For a given monomial order, any nonzero ideal has a unique reduced Gröbner basis.

Thus we can simply reduce bases to check if their ideals are equal.

The Ideal Membership Problem

When we combine the Gröbner bases with the divison algorithm, we have the ideal membership algorithm so that given an ideal I = \Braket{f_1, ..., f_s}, we can decide whether a given polynomial f lies in I as follows:

1. Find the Gröbner basis G = \Set{g_1, ..., g_t} for I, then
2. f \in I if and only if the remainder of f by G is 0.

The Problem of Solving Polynomial Equations

Assume we are given a system of equations such that

then these equations determine an ideal I = \Braket{f_1, ..., f_s} and we want to find all points \bold{V}(I).

Recall that \bold{V}(I) = \bold{V}(f_1,...,f_s). Now, if we use a Gröbner basis computations become suprisingly easy to solve. We will see this in more detail later on.

Check out the David Cox §8 for more examples in this regard.

The Implicitization Problem

Suppose

then just as above using Gröbner basis make computations much easier as we will see later on in more detail.

Check out the David Cox §8 for more examples in this regard.

Exercises

Def. \ell-th Elimination Ideal

Given I = \Braket{f_1, ..., f_s} \subseteq \mathbb{F}[x_1, ..., x_n], the \ell-th elimination ideal I_\ell is the ideal of \mathbb{F}[x_{\ell + 1},...,x_n] defined as

Thm. The Elimination Theorem

Let I \subseteq \mathbb{F}[x_1, ..., x_n] be an ideal and let G be a Gröbner basis of I with respect to lex order where x_1 > x_2 > \cdots > x_n, then for every 0 \leq \ell \leq n, the set

is a Gröbner basis of the \ell-th elimination ideal I_\ell.

The Elimination Theorem shows that a Gröbner basis for lex order eliminates not only the first variable, but also the first two variables, the first three variables, and so on.

Thm. The Extension Theorem

We saw that we can eliminate variables but now we will see that we can also extend our partial solutions to complete ones.

Let I = \Braket{f_1, ..., f_s} \subseteq \Complex[x_1, ..., x_n] and rewrite each f_i as

where T are the terms in which x_1 has degree less than N_i where N_i \geq 0 and 0 \neq g_i \in \Complex[x_2, ..., x_n].

Now, if \bold{V}(I_1) \ni (a_2, ..., a_n) \notin \bold{V}(g_1, ..., g_s), then there exists a_1 \in \Complex such that (a_1, ..., a_n) \in \bold{V}(I).

Notice how we stated the theorem for the field \Complex. Indeed, this theorem is false over \R.

We will later see that this and the following theorems are generally true for any algebraically closed field.

Corollary. Constant Coefficient Extension Theorem

The Extension Theorem becomes extra useful if one of the leading coefficients is consant.

Let I = \Braket{f_1, ..., f_s} \subseteq \Complex[x_1, ..., x_n] and assume for some f_i we have

where T are the terms in which x_1 has degree less than N where N > 0 and 0 \neq c \in \Complex[x_2, ..., x_n].

Now, if \bold{V}(I_1) \ni (a_2, ..., a_n), then there exists a_1 \in \Complex such that (a_1, ..., a_n) \in \bold{V}(I).

Direct result of the Extension Theorem.

Def. Projection Map

Let V = \mathbb{V}(f_1, ..., f_s) \subseteq \Complex^n, then we define the projection map

then, in \Complex^{n - \ell}, we have

This is basically the projection of V onto the last n - \ell components which is a subset of \bold{V}(I_\ell) — exercise.

Thm. Geometric Extesion Theorem

Let V = \bold{V}(f_1, ..., f_s) \subseteq \Complex^n and be g_i defined as in the extension theorem. For I = \Braket{f_1, ..., f_s}, we have the equality in \Complex^{n-1}

Proof follows from the extension theorem.

Thm. The Closure Theorem

Let I = \Braket{f_1, ..., f_s} and V = \bold{V}(I) \subseteq \Complex^n, then

1. \bold{V}(I_\ell) is the smallest* (affine) variety containing \pi_\ell(V) \subseteq \Complex^{n-\ell}, and
2. if V \neq \varnothing, then there exists an (affine) variety W \subset \bold{V}(I_\ell) such that \bold{V}(I_\ell) - W \subseteq \pi_\ell(V).

*smallest with respect to set inclusion.

Thm. Constant Coefficient Closure Theorem

Let I = \Braket{f_1, ..., f_s} and V = \bold{V}(I) \subseteq \Complex^n where some f_i is of the form

where T are the terms in which x_1 has degree less than N where N > 0 and 0 \neq c \in \Complex. Then, in \Complex^{n-1}

Def. Irreducible

A polynomial f \in \mathbb{F}[x_1, ..., x_n] is called irreducible over \mathbb{F} if f is non-constant and is not the product of two non-constant polynomials in \mathbb{F}[x_1, ..., x_n].

Thm. Division

Let f \in \mathbb{F}[x_1, ..., x_n] be an irreducible and suppose f divides the product gh, then f divides g or h.

Thm. Normal Form

Every non-constant f \in \mathbb{F}[x_1, ..., x_n] can be written uniquely as a product of irreducibles up to permutation.

Lemma. Common Factor Lemma

Let f, g \in \mathbb{F}[x] be polynomials of degree \ell > 0 and m > 0, respectively. Then f and g have a common factor if and only if there are polynomials A,B \in \mathbb{F}[x] such that:

1. A and B are not both zero.
2. A has degree at most m - 1 and B has degree at most \ell - 1.
3. Af + Bg = 0.

Def. Sylvester Matrix and Resultant

Given polynomials f, g \in \mathbb{F}[x] of positive degree, write them down in the form

where a_0 \neq 0 \neq b_0, then the Sylvester Matrix of f and g with respect to x, denoted \text{Syl}(f, g, x) is the (\ell + m) \times (\ell + m) coefficient matrix of the system of equations

so that

The resultant of f and g with respect to x denoted \text{Res}(f,g,x) is the determinant of the Sylvester Matrix:

Thm. Common Factor

Two polynomials f, g \in \mathbb{F}[x] of positive degree have a common factor in \mathbb{F}[x] if and only if \text{Res}(f,g,x) = 0.

Easy to prove using the lemma given above.

Def. Resultants in \mathbb{F}[x_1, ..., x_n]

Suppose f, g \in \mathbb{F}[x_1, ..., x_n] of positive degree in x_1, then to compute it's Sylvester Matrix, we write down

where a_i, b_i \in \mathbb{F}[x_2, ..., x_n], and we define the resultant of f and g with respect to x_1 to be the determinant with the corresponding coefficients

Thm. Resultant in Several Variables

Let f,g \in \mathbb{F}[x_1, ..., x_n] have positive degree in x_1, then

1. \text{Res}(f,g,x_1) is the first elimination ideal \Braket{f,g} \cap \mathbb{F}[x_2, ..., x_n].
2. \text{Res}(f,g,x_1) = 0 if and only if f and g have a common factor in \mathbb{F}[x_1, ..., x_n] which has positive degree in x_1.

Thm. The Weak Nullstellensatz

Let \mathbb{F} be an algebraically closed field and let I \subseteq \mathbb{F}[x_1, ..., x_n] be an ideal satisfying \bold{V}(I) = \varnothing, then I = \mathbb{F}[x_1, ..., x_n].

In the special case when \mathbb{F} = \Complex, the Weak Nullstellensatz may be thought of as the "Fundamental Theorem of Algebra for multivariable polynomials" so that every system of polynomials that generates an ideal strictly smaller than \Complex[x_1, ..., x_n] has a common zero in \Complex^n.

Thm. Consistency Algorithm

Suppose we have polynomials f_1, ..., f_s \in \Complex[x_1, ..., x_n], and we compute a reduced Gröbner basis B of the ideal \Braket{f_1, ..., f_s} with respect to any ordering. If this basis B is \{1\}, the polynomials have no common zero in \Complex^n, and if the basis is not \{1\}, they must have a common zero.

This arises from the fact that if 1 is an element of an ideal, then that ideal is the ring itself.

Thm. Hilbert's Nullstellensatz

Let \mathbb{F} be an algebraically closed field and f,f_1,...,f_s \in \mathbb{F}[x_1, ..., x_n] such that f \in \bold{I}(\bold{V}(f_1, ..., f_s)), then there exists an integer m \geq 1 such that

and the converse also holds.

Thm. Radical Ideal Lemma

Let V be a variety, then

Def. Radical

An ideal I is called radical if f^m \in I for some m \geq 1 implies that f \in I.

So \bold{I}(V) is always a radical ideal.

Def. Radical of an Ideal

Let I \subseteq \mathbb{F}[x_1,...,x_n] be an ideal, then the radical of I denoted \sqrt{I} is the set

Prove that \sqrt{I} is also an ideal and it's radical.

Thm. The Strong Nullstellensatz

This is usually the theorem referred to as The Nullstellensatz Theorem.

Let \mathbb{F} be an algebraically closed field and I an ideal in \mathbb{F}[x_1,...,x_n], then

Thm. Radical Membership

Let \mathbb{F} be an arbitrary field and let I = \Braket{f_1, ..., f_s} \subseteq \mathbb{F}[x_1,...,x_n] be an ideal, then f \in \sqrt{I} if and only if the constant polynomial 1 \in \tilde{I} = \Braket{f_1,...,f_s,1-yf} \subseteq \mathbb{F}[x_1,...,x_n,y].

Thm. Radicals and Irreducibles

Let f \in \mathbb{F}[x_1,...,x_n] and I = \Braket{f} be the principal ideal generated by f where f = cf_1^{a_1} \cdots f_r^{a_r} is the factorization of f into a product of distinct irreducible polynomials, then

Def. Reduced Polynomial

Suppose f \in \mathbb{F}[x_1, ..., x_n], then we define the reduction of f denoted f_\text{red} to be the polynomial such that

Moreover, we say a polynomial f is reduced or square-free if f = f_\text{red}.

So f_\text{red} is the polynomial with repeated factors "stripped away". Notice that f_\text{red} is unique up to a constant factor.

Thm. Computation of f_\text{red}

Suppose \mathbb{F} contains the field of rational numbers \mathbb{Q} and suppose I = \Braket{f} where f \in \mathbb{F}[x_1,...,x_n] then

Sums of Ideals

The sum of I and J is defined as the set

which is also an ideal in \mathbb{F}[x_1,...,x_n]. Moreover, I+J is the smallest ideal containing I and J so that

Therefore

Finally, adition of ideals corresponds geometrically to taking intersections of varieties, therefore

Product of Ideals

The product of ideals I and J is defined as the set

which is also an ideal in \mathbb{F}[x_1,...,x_n]. Moreover, I \cdot J is generated by the set of all products of generators of I and J

Finally, the product of ideals correspond geometrically to the operation of taking the unions of variaties

Thm. Intersection of Ideals

The intersection of two ideals I and J denoted I \cap J is also an ideal.

Moreover, intersection of two principal ideals is a principal ideal generated by their least common multiplier

Thm. Intersection Algorithm

Let I and J be ideals in \mathbb{F}[x_1,...,x_n], then

where (tI + (1-t)J) is an ideal in \mathbb{F}[x_1,...,x_n,t].

This theorem, with the combination of Gröbner basis and Elimination Theorem allows to compute a basis for the intersection of ideals.

Thm. \text{GCD} and \text{LCM}

Let f,g \in \mathbb{F}[x_1,...,x_n], then

Thm. Variety of Intersection

For any two ideals I and J in \mathbb{F}[x_1,...,x_n], we have

Thm. Intersection and Radical

Let I and J be ideals in \mathbb{F}[x_1,...,x_n], then