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 $I\subseteq R$ :

$I$ is generated by monomials,
If ${\textstyle f=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }x^{\alpha }\in I}$ , then $x^{\alpha }\in I$ , provided that $c_{\alpha }$ is nonzero.
$I$ 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 $m_{j}$ .^[1]

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 $f=f_{1}m_{1}+f_{2}m_{2}+\dotsm +f_{k}m_{k}$ , 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 $f=c_{1}m_{1}+c_{2}m_{2}+\dotsm +c_{k}m_{k}$ 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 $x^{a}y^{b}$ such that there exists $i$ such $a_{i}\leq a$ and $b_{i}\leq b.$ ^[2] 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 $\geq$ 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 $0$ .
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, $LT(I)=(LT(f)\mid f\in I)$ .
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 $LT(2x^{3}y+9xy^{5}+19)=2x^{3}y$ , but if we take y > x then $LT(2x^{3}y+9xy^{5}+19)=9xy^{5}$ .

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 $f=a_{1}g_{1}+a_{2}g_{2}+\dotsm +a_{s}g_{s}$ 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.

References

Miller, Ezra; Sturmfels, Bernd (2005), Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 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