Monomial ideal

Summary

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

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Let   be a field and   be the polynomial ring over   with n indeterminates  .

A monomial in   is a product   for an n-tuple   of nonnegative integers.

The following three conditions are equivalent for an ideal  :

  1.   is generated by monomials,
  2. If  , then  , provided that   is nonzero.
  3.   is torus fixed, i.e, given  , then   is fixed under the action   for all  .

We say that   is a monomial ideal if it satisfies any of these equivalent conditions.

Given a monomial ideal  ,   is in   if and only if every monomial ideal term   of   is a multiple of one the  .[1]

Proof: Suppose   and that   is in  . Then  , for some  .

For all  , we can express each   as the sum of monomials, so that   can be written as a sum of multiples of the  . Hence,   will be a sum of multiples of monomial terms for at least one of the  .

Conversely, let   and let each monomial term in   be a multiple of one of the   in  . Then each monomial term in   can be factored from each monomial in  . Hence   is of the form   for some  , as a result  .

The following illustrates an example of monomial and polynomial ideals.

Let   then the polynomial   is in I, since each term is a multiple of an element in J, i.e., they can be rewritten as   and   both in I. However, if  , then this polynomial   is not in J, since its terms are not multiples of elements in J.

Monomial ideals and Young diagrams

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Bivariate monomial ideals can be interpreted as Young diagrams.

Let   be a monomial ideal in   where   is a field. The ideal   has a unique minimal generating set of   of the form  , where   and  . The monomials in   are those monomials   such that there exists   such   and  [2] If a monomial   is represented by the point   in the plane, the figure formed by the monomials in   is often called the staircase of   because of its shape. In this figure, the minimal generators form the inner corners of a Young diagram.

The monomials not in   lie below the staircase, and form a vector space basis of the quotient ring  .

For example, consider the monomial ideal   The set of grid points   corresponds to the minimal monomial generators   Then as the figure shows, the pink Young diagram consists of the monomials that are not in  . The points in the inner corners of the Young diagram, allow us to identify the minimal monomials   in   as seen in the green boxes. Hence,  .

 
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   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  . Thus, monomial ideals can be described by Young diagrams of partitions.

Moreover, the  -action on the set of   such that   as a vector space over   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

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A monomial ordering is a well ordering   on the set of monomials such that if   are monomials, then  .

By the monomial order, we can state the following definitions for a polynomial in  .

Definition[1]

  1. Consider an ideal  , and a fixed monomial ordering. The leading term of a nonzero polynomial  , denoted by   is the monomial term of maximal order in   and the leading term of   is  .
  2. The ideal of leading terms, denoted by  , is the ideal generated by the leading terms of every element in the ideal, that is,  .
  3. A Gröbner basis for an ideal   is a finite set of generators   for   whose leading terms generate the ideal of all the leading terms in  , i.e.,   and  .

Note that   in general depends on the ordering used; for example, if we choose the lexicographical order on   subject to x > y, then  , but if we take y > x then  .

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  , the finite set of generators   is a Gröbner basis for  . To see this, note that any polynomial   can be expressed as   for  . Then the leading term of   is a multiple for some  . As a result,   is generated by the   likewise.

See also

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Footnotes

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References

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  • 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

Further reading

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  • Cox, David. "Lectures on toric varieties" (PDF). Lecture 3. §4 and §5.
  • 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 (PDF).