1.1 A brief introduction to computational algebraic geometry  (Page 8/20)

Over the next few weeks, we'll start learning about a computational tool called Gröbner bases that will tell us how to find such an $f$ , and will more generally allow us to study the question of, given polynomials $h_1,...h_k\in \mathbf{R}[x_1,...,x_n]$ , which polynomials $f\in \mathbf{R}[x_1,...,x_n]$ can be written in the form

for some $q_1,...,q_k\in \mathbf{R}[x_{,}...,x_n]$ ?

We'll begin with some terminology. If $f_1,...,f_k\in \mathbf{R}[x_1,...,x_n]$ are polynomials then the ideal generated by $f_1,...,f_k$ is the set

More generally, a non-empty subset $I\subseteq \mathbf{R}[x_1,...,x_n]$ is defined to be an ideal if $g_1{f}_1+g_2{f}_2\in I$ for every $f_1,f_2\in I$ and $g_1,g_2\in \mathbf{R}[x_1,...,x_n]$ .

Theorem (Hilbert Basis Theorem) Every ideal $I\subseteq R[x_1,...,x_n]$ is generated by finitely many polynomials, so that

for some $f_1,...,f_k\in I$ .

We probably won't prove this. We should note that this theorem doesn't say anything about the size of the smallest generating set of $I$ , so $k$ here could be much bigger than $n$ .

When dealing with polynomials in one variable, a polynomial always has a clear leading term, namely the term of highest degree. For polynomials in several variables, there are many different ways we might want to order the monomials. For convenience, if $\alpha =(a_1,...,a_n)$ is an $n$ -tuple of non-negative integers, then we will write

as an abbreviated notation for the corresponding monomial. Although there are many orderings on the monomials to choose from, we want them to respect the algebraic structure. For example if $x^{\alpha }$ divides $x^{\beta }$ , then we would like $x^{\alpha }$ to be smaller than $x^{\beta }$ .

A monomial order for $\mathbf{R}[x_1,...,x_n]$ is a total order This means that: (1) it is never the case that both $x^{\alpha }<x^{\beta }$ and $x^{\beta }<x^{\alpha }$ , and (2) if $x^{\alpha }<x^{\beta }$ and $x^{\beta }<x^{\gamma }$ , then $x^{\alpha }<x^{\gamma }$ . on the monomials such that if $x^{\alpha }<x^{\beta }$ then $x^{\gamma }{x}^{\alpha }<x^{\gamma }{x}^{\beta }$ for all monomials $x^{\gamma }$ which is a well-ordering Well-ordering means that if $S$ is any subset of monomials, then $S$ has a least element according to the ordering. This implies that 1 is the least monomial, since if $x^{\alpha }<1$ were the least monomial, then $x^{2\alpha }<x^{\alpha }$ would be even smaller, a contradiction. .

Example: Lexicographic order

Probably the simplest monomial ordering is the lexicographic (or “dictionary”) ordering. In this ordering, the power of the first variable is used to determine the order, with powers of the second variable only looked at when the first variable appears to the same power in two monomials. Similarly, we only look at the third variable when the first two are tied, and so on. For example, in the lex order for $\mathbf{R}[x,y,z]$ with $x>y>z$ , we have

More formally, given two monomials $x^{\alpha }$ and $x^{\beta }$ in $\mathbf{R}[x_1,...,x_n]$ , we say that $x^{\alpha }{>}_{\text{lex}}{x}^{\beta }$ if in the difference of vectors $\alpha -\beta$ , the leftmost non-zero entry is positive. One can check that this does in fact define a monomial order. See section 2.2 of Cox, Little, and O'Shea for more details about term orderings, including proofs that the well-ordering property holds, etc.

Example: Graded lexicographic order

One thing we might not like about lex order is that it doesn't respect degrees (e.g. $xy>y^3{z}^4$ ). We can define a new order, called graded lexicographic order by saying that higher degree monomials are bigger and using lex order to break ties. For example,