Monomial ideal

Let ${\displaystyle \mathbb {K} }$ be a field and ${\displaystyle R=\mathbb {K} [x]}$ be the polynomial ring over ${\displaystyle \mathbb {K} }$ with n variables ${\displaystyle x=x_{1},x_{2},\dotsc ,x_{n}}$.

A monomial in ${\displaystyle R}$ is a product ${\displaystyle x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{n}^{\alpha _{n}}}$ for an n-tuple ${\displaystyle \alpha =(\alpha _{1},\alpha _{2},\dotsc ,\alpha _{n})\in \mathbb {N} ^{n}}$ of nonnegative integers.

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

1. ${\displaystyle I}$ is generated by monomials,
2. If ${\textstyle f=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }x^{\alpha }\in I}$, then ${\displaystyle x^{\alpha }\in I}$, provided that ${\displaystyle c_{\alpha }}$ is nonzero.
3. ${\displaystyle I}$ is torus fixed, i.e, given ${\displaystyle (c_{1},c_{2},\dotsc ,c_{n})\in (\mathbb {K} ^{*})^{n}}$, then ${\displaystyle I}$ is fixed under the action ${\displaystyle f(x_{i})=c_{i}x_{i}}$ for all ${\displaystyle i}$.

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

Given a monomial ideal ${\displaystyle I=(m_{1},m_{2},\dotsc ,m_{k})}$, ${\displaystyle f\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]}$ is in ${\displaystyle I}$ if and only if every monomial ideal term ${\displaystyle f_{i}}$ of ${\displaystyle f}$ is a multiple of one the ${\displaystyle m_{j}}$.[1]

Proof: Suppose ${\displaystyle I=(m_{1},m_{2},\dotsc ,m_{k})}$ and that ${\displaystyle f\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]}$ is in ${\displaystyle I}$. Then ${\displaystyle f=f_{1}m_{1}+f_{2}m_{2}+\dotsm +f_{k}m_{k}}$, for some ${\displaystyle f_{i}\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]}$.

For all ${\displaystyle 1\leqslant i\leqslant k}$, we can express each ${\displaystyle f_{i}}$ as the sum of monomials, so that ${\displaystyle f}$ can be written as a sum of multiples of the ${\displaystyle m_{i}}$. Hence, ${\displaystyle f}$ will be a sum of multiples of monomial terms for at least one of the ${\displaystyle m_{i}}$.

Conversely, let ${\displaystyle I=(m_{1},m_{2},\dotsc ,m_{k})}$ and let each monomial term in ${\displaystyle f\in \mathbb {K} [x_{1},x_{2},...,x_{n}]}$ be a multiple of one of the ${\displaystyle m_{i}}$ in ${\displaystyle I}$. Then each monomial term in ${\displaystyle I}$ can be factored from each monomial in ${\displaystyle f}$. Hence ${\displaystyle f}$ is of the form ${\displaystyle f=c_{1}m_{1}+c_{2}m_{2}+\dotsm +c_{k}m_{k}}$ for some ${\displaystyle c_{i}\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]}$, as a result ${\displaystyle f\in I}$.

The following illustrates an example of monomial and polynomial ideals.

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

A monomial ideal can be interpreted as a Young diagram. Suppose ${\displaystyle I\in \mathbb {R} [x,y]}$, then ${\displaystyle I}$ can be interpreted in terms of the minimal monomials generators as ${\displaystyle I=(x^{a_{1}}y^{b_{1}},x^{a_{2}}y^{b_{2}},\dotsc ,x^{a_{k}}y^{b_{k}})}$, where ${\displaystyle a_{1}>a_{2}>\dotsm >a_{k}\geq 0}$ and ${\displaystyle b_{k}>\dotsm >b_{2}>b_{1}\geq 0}$. The minimal monomial generators of ${\displaystyle I}$ can be seen as the inner corners of the Young diagram. The minimal generators would determine where we would draw the staircase diagram.[2] The monomials not in ${\displaystyle I}$ lie inside the staircase, and these monomials form a vector space basis for the quotient ring ${\displaystyle \mathbb {R} [x,y]/I}$.

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

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 ${\displaystyle (a_{i},b_{j})}$ 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 ${\displaystyle I}$. Thus, monomial ideals can be described by Young diagrams of partitions.

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

A monomial ordering is a well ordering ${\displaystyle \geq }$ on the set of monomials such that if ${\displaystyle a,m_{1},m_{2}}$ are monomials, then ${\displaystyle am_{1}\geq am_{2}}$.

By the monomial order, we can state the following definitions for a polynomial in ${\displaystyle \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]}$.

Definition[1]

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

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

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

Notice that for a monomial ideal ${\displaystyle I=(g_{1},g_{2},\dotsc ,g_{s})\in \mathbb {F} [x_{1},x_{2},\dotsc ,x_{n}]}$, the finite set of generators ${\displaystyle {\{g_{1},g_{2},\dotsc ,g_{s}}\}}$ is a Gröbner basis for ${\displaystyle I}$. To see this, note that any polynomial ${\displaystyle f\in I}$ can be expressed as ${\displaystyle f=a_{1}g_{1}+a_{2}g_{2}+\dotsm +a_{s}g_{s}}$ for ${\displaystyle a_{i}\in \mathbb {F} [x_{1},x_{2},\dotsc ,x_{n}]}$. Then the leading term of ${\displaystyle f}$ is a multiple for some ${\displaystyle g_{i}}$. As a result, ${\displaystyle LT(I)}$ is generated by the ${\displaystyle g_{i}}$ likewise.

• Stanley–Reisner ring
• Hodge algebra

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