Toric 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 $\mathbb$ be a field and R = \mathbb /math> be the polynomial ring over $\mathbb$ with ''n'' variables $x = x_1, x_2, \dotsc, x_n$.

A monomial in $R$ is a product $x^ = x_1^ x_2^ \cdots x_n^$ for an ''n''-tuple $\alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb^n$ of nonnegative integers.

The following three conditions are equivalent for an ideal $I \subseteq R$:
# $I$ is generated by monomials,
# If $f = \sum_ c_ x^ \in I$, then $x^ \in I$, provided that $c_$ is nonzero.
# $I$ is torus fixed, i.e, given $(c_1, c_2, \dotsc, c_n) \in (\mathbb^*)^$, then $I$ is fixed under the action $f(x_i) = c_ix_i$ for all $i$.

We say that I \subseteq \mathbb /math> 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 _1, x_2, \dotsc, x_n/math> 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 _1, x_2, \dotsc, x_n/math> is in $I$. Then $f = f_1m_1 + f_2m_2 + \dotsm + f_km_k$, for some f_i \in \mathbb _1, x_2, \dotsc, x_n/math>.

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 _1, x_2, . . . , x_n/math> 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 _1, x_2, \dotsc, x_n/math>, as a result $f \in I$.

The following illustrates an example of monomial and polynomial ideals.

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

Monomial Ideals and Young Diagrams

A monomial ideal can be interpreted as a Young diagram. Suppose $I \in \mathbb$, y/math>, then $I$ can be interpreted in terms of the minimal monomials generators as $I = (x^y^, x^y^,\dotsc, x^y^)$, where $a_1 > a_2 > \dotsm > a_k \geq 0$ and $b_k > \dotsm > b_2 > b_1 \geq 0$. The minimal monomial generators of $I$ 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 $I$ lie inside the staircase, and these monomials form a vector space basis for the quotient ring $\mathbb$, yI.

Consider the following example.
Let $I = (x^3, x^2y, y^3) \subset \mathbb$, y/math> be a monomial ideal. Then the set of grid points $S = \subset \mathbb^2$ corresponds to the minimal monomial generators $x^3y^0, x^2y^1, x^0y^3$ in $I$. 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 $(\mathbb^*)^2$-action on the set of $I \subset \mathbb$, y/math> such that $\dim_ \mathbb$, yI = n as a vector space over $\mathbb$ 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 $\geq$ 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 _1, x_2, \dotsc, x_n.

''Definition''

# Consider an ideal I \subset \mathbb _1, x_2, \dotsc, x_n, and a fixed monomial ordering. The leading term of a nonzero polynomial f \in \mathbb _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 _1, x_2, \dotsc, x_n is a finite set of generators $$ 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 $\mathbb$, y/math> 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 with several variables.

Notice that for a monomial ideal I = (g_1, g_2, \dotsc, g_s) \in \mathbb _1, x_2, \dotsc, x_n/math>, the finite set of generators $$ 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 _1, x_2, \dotsc, x_n/math>. 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.

See also

*Stanley–Reisner ring
* Hodge algebra

Footnotes

References

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Further reading

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*{{cite book, first=Bernard, last=Teissier, authorlink=Bernard Teissier, url=http://library.msri.org/books/Book51/files/07teissier.pdf, title= Monomial Ideals, Binomial Ideals, Polynomial Ideals, year= 2004

Homogeneous polynomials
Algebra