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 a prime ideal). 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 [math]\displaystyle{ \mathbb{K} }[/math] be a field and [math]\displaystyle{ R = \mathbb{K}[x] }[/math] be the polynomial ring over [math]\displaystyle{ \mathbb{K} }[/math] with n variables [math]\displaystyle{ x = x_1, x_2, \dotsc, x_n }[/math].

A monomial in [math]\displaystyle{ R }[/math] is a product [math]\displaystyle{ x^{\alpha} = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n} }[/math] for an n-tuple [math]\displaystyle{ \alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb{N}^n }[/math] of nonnegative integers.

The following three conditions are equivalent for an ideal [math]\displaystyle{ I \subseteq R }[/math]:

1. [math]\displaystyle{ I }[/math] is generated by monomials,
2. If [math]\displaystyle{ f = \sum_{\alpha \in \mathbb{N}^n} c_{\alpha} x^{\alpha} \in I }[/math], then [math]\displaystyle{ x^{\alpha} \in I }[/math], provided that [math]\displaystyle{ c_{\alpha} }[/math] is nonzero.
3. [math]\displaystyle{ I }[/math] is torus fixed, i.e, given [math]\displaystyle{ (c_1, c_2, \dotsc, c_n) \in (\mathbb{K}^*)^{n} }[/math], then [math]\displaystyle{ I }[/math] is fixed under the action [math]\displaystyle{ f(x_i) = c_ix_i }[/math] for all [math]\displaystyle{ i }[/math].

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

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

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

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

Conversely, let [math]\displaystyle{ I = (m_1, m_2, \dotsc, m_k) }[/math] and let each monomial term in [math]\displaystyle{ f \in \mathbb{K} [x_1, x_2, . . . , x_n] }[/math] be a multiple of one of the [math]\displaystyle{ m_i }[/math] in [math]\displaystyle{ I }[/math]. Then each monomial term in [math]\displaystyle{ I }[/math] can be factored from each monomial in [math]\displaystyle{ f }[/math]. Hence [math]\displaystyle{ f }[/math] is of the form [math]\displaystyle{ f = c_1m_1 + c_2m_2 + \dotsm + c_km_k }[/math] for some [math]\displaystyle{ c_i \in \mathbb{K}[x_1, x_2, \dotsc, x_n] }[/math], as a result [math]\displaystyle{ f \in I }[/math].

The following illustrates an example of monomial and polynomial ideals.

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

Monomial Ideals and Young Diagrams

A monomial ideal can be interpreted as a Young diagram. Suppose [math]\displaystyle{ I \in \mathbb{R}[x, y] }[/math], then [math]\displaystyle{ I }[/math] can be interpreted in terms of the minimal monomials generators as [math]\displaystyle{ I = (x^{a_1}y^{b_1}, x^{a_2}y^{b_2},\dotsc, x^{a_k}y^{b_k}) }[/math], where [math]\displaystyle{ a_1 \gt a_2 \gt \dotsm \gt a_k \geq 0 }[/math] and [math]\displaystyle{ b_k \gt \dotsm \gt b_2 \gt b_1 \geq 0 }[/math]. The minimal monomial generators of [math]\displaystyle{ I }[/math] can be seen as the inner corners of the Young diagram. The minimal generators would determine where we would draw the staircase diagram. The monomials not in [math]\displaystyle{ I }[/math] lie inside the staircase, and these monomials form a vector space basis for the quotient ring [math]\displaystyle{ \mathbb{R}[x, y]/I }[/math].

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

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 [math]\displaystyle{ (a_i, b_j) }[/math] 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 [math]\displaystyle{ I }[/math]. Thus, monomial ideals can be described by Young diagrams of partitions.

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

Monomial Ordering and Gröbner Basis

A monomial ordering is a well ordering [math]\displaystyle{ \geq }[/math] on the set of monomials such that if [math]\displaystyle{ a, m_1, m_2 }[/math] are monomials, then [math]\displaystyle{ am_1 \geq am_2 }[/math].

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

Definition

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

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

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 [math]\displaystyle{ I = (g_1, g_2, \dotsc, g_s) \in \mathbb{F}[x_1, x_2, \dotsc, x_n] }[/math], the finite set of generators [math]\displaystyle{ {\{g_1, g_2, \dotsc, g_s}\} }[/math] is a Gröbner basis for [math]\displaystyle{ I }[/math]. To see this, note that any polynomial [math]\displaystyle{ f \in I }[/math] can be expressed as [math]\displaystyle{ f = a_1g_1 + a_2g_2 + \dotsm + a_sg_s }[/math] for [math]\displaystyle{ a_i \in \mathbb{F}[x_1, x_2, \dotsc, x_n] }[/math]. Then the leading term of [math]\displaystyle{ f }[/math] is a multiple for some [math]\displaystyle{ g_i }[/math]. As a result, [math]\displaystyle{ LT(I) }[/math] is generated by the [math]\displaystyle{ g_i }[/math] likewise.

See also

• Stanley–Reisner ring
• Hodge algebra

References

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