##
**Binomial ideals.**
*(English)*
Zbl 0873.13021

This work is a systematic study of ideals of the polynomial ring \(k[x_1, x_2, \dots, x_n]\) (over a field \(k)\), which are generated by binomials \(ax^{\alpha_1}_1 x_2^{ \alpha_2} \dots x_n^{\alpha_n} +bx_1^{\beta_1} x_2^{\beta_2} \dots x_n^{\beta_n}\). These ideals arose in recent years in several contexts of commutative algebra and algebraic geometry (e.g., the important class of toric ideals and corresponding varieties, commutative semigroup algebras, Stanley’s face rings of polyhedral complexes), relevant questions of numerical mathematics and computational aspects of some applied problems (e.g., in the theory of dynamical systems, computational statistics, computer algebra). Starting from the fact that the reduced Gröbner basis of a binomial ideal consists of binomials, the authors obtain many corollaries concerning the ideals generated by binomials and monomials. For instance, the quotient of a binomial ideal by a single monomial is a binomial ideal, but this is generally not true for the quotient of a binomial ideal by a monomial ideal.

Further, binomial ideals in the ring of Laurent polynomials \(k[x_1,x_2, \dots, x_n,x_1^{-1},\;x_2^{-1}, \dots, x_n^{-1}]\) are described using partial characters on the lattice of monomials, i.e., group homomorphisms from a subgroup \(L\) of this lattice to the multiplicative group \(k^*\) of \(k\). As one of the corollaries from characterization of algebraic sets, a condition for a binomial ideal in \(k[x_1,x_2, \dots, x_n]\) to be prime is obtained provided \(k\) is algebraically closed. In this case, binomial prime ideals are the same as toric ideals. – It is shown that the ordinary radical and \(k\)-radical of a binomial ideal are binomial, as well. Finally, it is proved that (in the case of an algebraically closed field \(k)\) any binomial ideal in \(k[x_1, x_2, \dots, x_n]\) has a minimal primary decomposition in terms of binomial ideals. As a preliminary, the notion of a cellular ideal is introduced as such ideal \(I\) that for some \({\mathcal E} \subseteq \{1, \dots, n\}\), one has \(I=(I: (\prod_{i\in {\mathcal E}} x_i)^\infty)\) and \(I\) contains a power of \(M({\mathcal E}) =(\{x_i\}_{i\notin {\mathcal E}})\); a decomposition into cellular binomial ideals is obtained. – In addition, certain cases, where the cellular decomposition is already a primary decomposition are pointed out.

The exposition is illustrated by many examples. The corresponding algorithms are formulated.

Further, binomial ideals in the ring of Laurent polynomials \(k[x_1,x_2, \dots, x_n,x_1^{-1},\;x_2^{-1}, \dots, x_n^{-1}]\) are described using partial characters on the lattice of monomials, i.e., group homomorphisms from a subgroup \(L\) of this lattice to the multiplicative group \(k^*\) of \(k\). As one of the corollaries from characterization of algebraic sets, a condition for a binomial ideal in \(k[x_1,x_2, \dots, x_n]\) to be prime is obtained provided \(k\) is algebraically closed. In this case, binomial prime ideals are the same as toric ideals. – It is shown that the ordinary radical and \(k\)-radical of a binomial ideal are binomial, as well. Finally, it is proved that (in the case of an algebraically closed field \(k)\) any binomial ideal in \(k[x_1, x_2, \dots, x_n]\) has a minimal primary decomposition in terms of binomial ideals. As a preliminary, the notion of a cellular ideal is introduced as such ideal \(I\) that for some \({\mathcal E} \subseteq \{1, \dots, n\}\), one has \(I=(I: (\prod_{i\in {\mathcal E}} x_i)^\infty)\) and \(I\) contains a power of \(M({\mathcal E}) =(\{x_i\}_{i\notin {\mathcal E}})\); a decomposition into cellular binomial ideals is obtained. – In addition, certain cases, where the cellular decomposition is already a primary decomposition are pointed out.

The exposition is illustrated by many examples. The corresponding algorithms are formulated.

Reviewer: J.Rosenknop (Berlin)

### MSC:

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13F25 | Formal power series rings |

### Keywords:

ideals of polynomial ring; binomials; Gröbner basis; Laurent polynomials; lattice of monomials; cellular binomial ideals
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\textit{D. Eisenbud} and \textit{B. Sturmfels}, Duke Math. J. 84, No. 1, 1--45 (1996; Zbl 0873.13021)

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