Monomial ideal

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

A toric ideal is an ideal generated by differences of monomials (provided the ideal is prime). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal.

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

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 $$\alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb{N}^n$$ of nonnegative integers.

The following three conditions are equivalent for an ideal $$I \subseteq R$$:
 * 1) $$I$$ is generated by monomials,
 * 2) If $f = \sum_{\alpha \in \mathbb{N}^n} c_{\alpha} x^{\alpha} \in I$, then $$x^{\alpha} \in I$$, provided that $$c_{\alpha}$$ is nonzero.
 * 3) $$I$$ is torus fixed, i.e, given $$(c_1, c_2, \dotsc, c_n) \in (\mathbb{K}^*)^n$$, 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 $$I = (m_1, m_2, \dotsc, m_k)$$, $$f \in \mathbb{K}[x_1, x_2, \dotsc, x_n]$$ is in $$I$$ if and only if every monomial ideal term $$f_i$$ of $$f$$ is a multiple of one the $$m_j$$.

Proof: Suppose $$I = (m_1, m_2, \dotsc, m_k)$$ and that $$f \in \mathbb{K}[x_1, x_2, \dotsc, x_n]$$ is in $$I$$. Then $$f = f_1m_1 + f_2m_2 + \dotsm + f_km_k$$, for some $$f_i \in \mathbb{K}[x_1, x_2, \dotsc, x_n]$$.

For all $$1 \leqslant i \leqslant k$$, 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 $$I = (m_1, m_2, \dotsc, m_k)$$ 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 $$f = c_1m_1 + c_2m_2 + \dotsm + c_km_k$$ for some $$c_i \in \mathbb{K}[x_1, x_2, \dotsc, x_n]$$, as a result $$f \in I$$.

The following illustrates an example of monomial and polynomial ideals.

Let $$I = (xyz, y^2)$$ then the polynomial $$x^2 yz + 3xy^2$$ 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 $$3xy^2 = 3x(y^2),$$ both in $I$. However, if $$J = (xz^2, y^2)$$, then this polynomial $$x^2 yz + 3xy^2$$ 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 $$\{x^{a_1}y^{b_1}, x^{a_2}y^{b_2},\ldots, x^{a_k}y^{b_k}\}$$, where $$a_1 > a_2 > \dotsm > a_k \geq 0$$ and $$b_k > \dotsm > b_2 > b_1 \geq 0$$. The monomials in $$I$$ are those monomials $$x^ay^b$$ such that there exists $$i$$ such $$a_i\le a$$ and $$b_i\le b.$$ If a monomial $$x^ay^b$$ 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 $$I = (x^3, x^2y, y^3) \subset k[x, y].$$ The set of grid points $$S = {\{(3, 0), (2, 1),(0, 3)}\}$$ corresponds to the minimal monomial generators $$x^3y^0, x^2y^1, x^0y^3.$$ 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 $$x^0y^3, x^2y^1, x^3y^0$$ in $$I$$ as seen in the green boxes. Hence, $$I = (y^3, x^2y, x^3)$$.

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 $$\geq$$ on the set of monomials such that if $$a, m_1, m_2$$ are monomials, then $$am_1 \geq am_2$$.

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

Definition


 * 1) Consider an ideal $$I \subset \mathbb{K}[x_1, x_2, \dotsc, x_n]$$, and a fixed monomial ordering. The leading term of a nonzero polynomial $$f \in \mathbb{K}[x_1, x_2, \dotsc, x_n]$$, 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 $$I \subset \mathbb{K}[x_1, x_2, \dotsc, x_n]$$ is a finite set of generators $${\{g_1, g_2, \dotsc, g_s}\}$$ for $$I$$ whose leading terms generate the ideal of all the leading terms in $$I$$, i.e., $$I = (g_1, g_2, \dotsc, g_s)$$ and $$LT(I) = (LT(g_1), LT(g_2), \dotsc, LT(g_s))$$.

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 $$LT(2x^3y + 9xy^5 + 19) = 2x^3y$$, but if we take y > x then $$LT(2x^3y + 9xy^5 + 19) = 9xy^5$$.

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 $$I = (g_1, g_2, \dotsc, g_s) \in \mathbb{F}[x_1, x_2, \dotsc, x_n]$$, the finite set of generators $${\{g_1, g_2, \dotsc, g_s}\}$$ is a Gröbner basis for $$I$$. To see this, note that any polynomial $$f \in I$$ can be expressed as $$f = a_1g_1 + a_2g_2 + \dotsm + a_sg_s$$ for $$a_i \in \mathbb{F}[x_1, x_2, \dotsc, x_n]$$. 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.