Monomial ideal

Short description: Ideal generated by one-term polynomials

In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field.

Definitions and properties

Let $\mathbb{𝕂}$ be a field and $R=\mathbb{𝕂}[x]$ be the polynomial ring over $\mathbb{𝕂}$ with n indeterminates $x=x_1,x_2,\dots ,x_n$.

A monomial in $R$ is a product $x^{\alpha }=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\cdots x_n^{{\alpha }_n}$ for an n-tuple $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\in {\mathbb{\mathbb{N}}}^n$ of nonnegative integers.

The following three conditions are equivalent for an ideal $I\subseteq R$:

1. $I$ is generated by monomials,
2. If $f=\sum _{\alpha \in {\mathbb{\mathbb{N}}}^n}{c}_{\alpha }{x}^{\alpha }\in I$, then $x^{\alpha }\in I$, provided that $c_{\alpha }$ is nonzero.
3. $I$ is torus fixed, i.e, given $(c_1,c_2,\dots ,c_n)\in ({\mathbb{𝕂}}^{*}{)}^n$, then $I$ is fixed under the action $f(x_i)=c_i{x}_i$ for all $i$.

We say that $I\subseteq \mathbb{𝕂}[x]$ is a monomial ideal if it satisfies any of these equivalent conditions.

Given a monomial ideal $I=(m_1,m_2,\dots ,m_k)$, $f\in \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$ is in $I$ if and only if every monomial term $f_i$ of $f$ is a multiple of one of the $m_j$.

Proof: Suppose $I=(m_1,m_2,\dots ,m_k)$ and that $f\in \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$ is in $I$. Then $f=f_1{m}_1+f_2{m}_2+\cdots +f_k{m}_k$, for some $f_i\in \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$.

For all $1⩽i⩽k$, we can express each $f_i$ as the sum of monomials, so that $f$ can be written as a sum of multiples of the $m_i$. Hence, $f$ will be a sum of multiples of monomial terms for at least one of the $m_i$.

Conversely, let $I=(m_1,m_2,\dots ,m_k)$ and let each monomial term in $f\in \mathbb{𝕂}[x_1,x_2,...,x_n]$ be a multiple of one of the $m_i$ in $I$. Then each monomial term in $I$ can be factored from each monomial in $f$. Hence $f$ is of the form $f=c_1{m}_1+c_2{m}_2+\cdots +c_k{m}_k$ for some $c_i\in \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$, as a result $f\in I$.

The following illustrates an example of monomial and polynomial ideals.

Let $I=(xyz,y^2)$ then the polynomial $x^{2}yz+3x{y}^2$ is in I, since each term is a multiple of an element in J, i.e., they can be rewritten as $x^{2}yz=x(xyz)$ and $3x{y}^2=3x(y^2),$ both in I. However, if $J=(x{z}^2,y^2)$, then this polynomial $x^{2}yz+3x{y}^2$ is not in J, since its terms are not multiples of elements in J.

Monomial ideals and Young diagrams

Bivariate monomial ideals can be interpreted as Young diagrams.

Let $I$ be a monomial ideal in $I\subset k[x,y],$ where $k$ is a field. The ideal $I$ has a unique minimal generating set of $I$ of the form $\{x^{a_1}{y}^{b_1},x^{a_2}{y}^{b_2},\dots ,x^{a_k}{y}^{b_k}\}$, where $a_1>a_2>\cdots >a_k\ge 0$ and $b_k>\cdots >b_2>b_1\ge 0$. The monomials in $I$ are those monomials $x^a{y}^b$ such that there exists $i$ such $a_i\le a$ and $b_i\le b.$ If a monomial $x^a{y}^b$ is represented by the point $(a,b)$ in the plane, the figure formed by the monomials in $I$ is often called the staircase of $I,$ because of its shape. In this figure, the minimal generators form the inner corners of a Young diagram.

The monomials not in $I$ lie below the staircase, and form a vector space basis of the quotient ring $k[x,y]/I$.

For example, consider the monomial ideal $I=(x^3,x^{2}y,y^3)\subset k[x,y].$ The set of grid points $S=\{(3,0),(2,1),(0,3)\}$ corresponds to the minimal monomial generators $x^3{y}^0,x^2{y}^1,x^0{y}^3.$ Then as the figure shows, the pink Young diagram consists of the monomials that are not in $I$. The points in the inner corners of the Young diagram, allow us to identify the minimal monomials $x^0{y}^3,x^2{y}^1,x^3{y}^0$ in $I$ as seen in the green boxes. Hence, $I=(y^3,x^{2}y,x^3)$.

A Young diagram and its connection with its monomial ideal.

In general, to any set of grid points, we can associate a Young diagram, so that the monomial ideal is constructed by determining the inner corners that make up the staircase diagram; likewise, given a monomial ideal, we can make up the Young diagram by looking at the $(a_i,b_j)$ and representing them as the inner corners of the Young diagram. The coordinates of the inner corners would represent the powers of the minimal monomials in $I$. Thus, monomial ideals can be described by Young diagrams of partitions.

Moreover, the $({\mathbb{ℂ}}^{*}{)}^2$-action on the set of $I\subset \mathbb{ℂ}[x,y]$ such that ${\dim }_{\mathbb{ℂ}}\mathbb{ℂ}[x,y]/I=n$ as a vector space over $\mathbb{ℂ}$ has fixed points corresponding to monomial ideals only, which correspond to integer partitions of size n, which are identified by Young diagrams with n boxes.

Monomial orderings and Gröbner bases

A monomial ordering is a well ordering $\ge$ on the set of monomials such that if $a,m_1,m_2$ are monomials, then $a{m}_1\ge a{m}_2$.

By the monomial order, we can state the following definitions for a polynomial in $\mathbb{𝕂}[x_1,x_2,\dots ,x_n]$.

Definition

1. Consider an ideal $I\subset \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$, and a fixed monomial ordering. The leading term of a nonzero polynomial $f\in \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$, denoted by $LT(f)$ is the monomial term of maximal order in $f$ and the leading term of $f=0$ is $0$.
2. The ideal of leading terms, denoted by $LT(I)$, is the ideal generated by the leading terms of every element in the ideal, that is, $LT(I)=(LT(f)\mid f\in I)$.
3. A Gröbner basis for an ideal $I\subset \mathbb{𝕂}[x_1,x_2,\dots ,x_n]$ is a finite set of generators $\{g_1,g_2,\dots ,g_s\}$ for $I$ whose leading terms generate the ideal of all the leading terms in $I$, i.e., $I=(g_1,g_2,\dots ,g_s)$ and $LT(I)=(LT(g_1),LT(g_2),\dots ,LT(g_s))$.

Note that $LT(I)$ in general depends on the ordering used; for example, if we choose the lexicographical order on $\mathbb{ℝ}[x,y]$ subject to x > y, then $LT(2x^{3}y+9x{y}^5+19)=2x^{3}y$, but if we take y > x then $LT(2x^{3}y+9x{y}^5+19)=9x{y}^5$.

In addition, monomials are present on Gröbner basis and to define the division algorithm for polynomials in several indeterminates.

Notice that for a monomial ideal $I=(g_1,g_2,\dots ,g_s)\in \mathbb{𝔽}[x_1,x_2,\dots ,x_n]$, the finite set of generators $\{g_1,g_2,\dots ,g_s\}$ is a Gröbner basis for $I$. To see this, note that any polynomial $f\in I$ can be expressed as $f=a_1{g}_1+a_2{g}_2+\cdots +a_s{g}_s$ for $a_i\in \mathbb{𝔽}[x_1,x_2,\dots ,x_n]$. Then the leading term of $f$ is a multiple for some $g_i$. As a result, $LT(I)$ is generated by the $g_i$ likewise.

Alternatively, this follows immediately from Buchberger's Criterion, since the S-polynomial of any two monomials is $0$ by definition.

See also

• Stanley–Reisner ring
• Hodge algebra

References

• Miller, Ezra; Sturmfels, Bernd (2005), Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, New York: Springer-Verlag, ISBN 0-387-22356-8 
• Dummit, David S.; Foote, Richard M. (2004), Abstract Algebra (third ed.), New York: John Wiley & Sons, ISBN 978-0-471-43334-7 

Further reading

• Cox, David. "Lectures on toric varieties". Lecture 3. §4 and §5. https://dacox.people.amherst.edu/lectures/coxcimpa.pdf. 
• Sturmfels, Bernd (1996). Gröbner Bases and Convex Polytopes. Providence, RI: American Mathematical Society. 
• Taylor, Diana Kahn (1966). Ideals generated by monomials in an R-sequence (PhD thesis). University of Chicago. MR 2611561. ProQuest 302227382.
• Teissier, Bernard (2004). Monomial Ideals, Binomial Ideals, Polynomial Ideals. http://library.msri.org/books/Book51/files/07teissier.pdf.