# zbMATH — the first resource for mathematics

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.

##### 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
Full Text:
##### References:
 [1] V. I. Arnold, $$A$$-graded algebras and continued fractions , Comm. Pure Appl. Math. 42 (1989), no. 7, 993-1000. · Zbl 0692.16012 · doi:10.1002/cpa.3160420705 [2] D. Bayer and M. Stillman, On the complexity of computing syzygies , J. Symbolic Comput. 6 (1988), no. 2-3, 135-147. · Zbl 0667.68053 · doi:10.1016/S0747-7171(88)80039-7 [3] E. Becker, R. Grobe, and M. Niermann, Real zeros and real radicals of binomial ideals , manuscript, 1996. · Zbl 0889.13001 · doi:10.1016/S0022-4049(97)00004-2 [4] W. D. Brownawell, Bounds for the degrees in the Nullstellensatz , Ann. of Math. (2) 126 (1987), no. 3, 577-591. JSTOR: · Zbl 0641.14001 · doi:10.2307/1971361 · links.jstor.org [5] B. Buchberger, Gröbner bases: an algorithmic method in polynomial ideal theory , Multidimensional Systems Theory, Math. Appl., vol. 16, D. Reidel, Dordrecht, 1985. · Zbl 0587.13009 [6] D. Buchsbaum and D. Eisenbud, Generic free resolutions and a family of generically perfect ideals , Advances in Math. 18 (1975), no. 3, 245-301. · Zbl 0336.13007 · doi:10.1016/0001-8708(75)90046-8 [7] D. Buchsbaum and D. S. Rim, A generalized Koszul complex. II. Depth and multiplicity , Trans. Amer. Math. Soc. 111 (1964), 197-224. JSTOR: · Zbl 0131.27802 · doi:10.2307/1994241 · links.jstor.org [8] P. Conti and C. Traverso, Buchberger algorithm and integer programming , Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991), Lecture Notes in Comput. Sci., vol. 539, Springer, Berlin, 1991, pp. 130-139. · Zbl 0771.13014 [9] D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms , Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. · Zbl 0756.13017 [10] W. Decker, N. Manolache, and F. Schreyer, Geometry of the Horrocks bundle on $$\mathbf P^ 5$$ , Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 128-148. · Zbl 0774.14013 [11] P. Diaconis and B. Sturmfels, Algebraic algorithms for generating from conditional distributions , to appear in Ann. Statist. · Zbl 0952.62088 · doi:10.1214/aos/1030563990 [12] D. Eisenbud, Commutative algebra with a View Toward Algebraic Geometry , Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. · Zbl 0819.13001 [13] D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct methods for primary decomposition , Invent. Math. 110 (1992), no. 2, 207-235. · Zbl 0770.13018 · doi:10.1007/BF01231331 · eudml:144049 [14] D. Eisenbud and B. Sturmfels, Finding sparse systems of parameters , J. Pure Appl. Algebra 94 (1994), no. 2, 143-157. · Zbl 0807.13012 · doi:10.1016/0022-4049(94)90029-9 [15] W. Fulton, Introduction to toric varieties , Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. · Zbl 0813.14039 [16] P. Gianni, B. Trager, and G. Zacharias, Gröbner bases and primary decomposition of polynomial ideals , J. Symbolic Comput. 6 (1988), no. 2-3, 149-167. · Zbl 0667.13008 · doi:10.1016/S0747-7171(88)80040-3 [17] R. Gilmer, Commutative semigroup rings , Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1984. · Zbl 0566.20050 [18] I. Hoveijn, Aspects of resonance in dynamical systems , Ph.D. thesis, University of Utrecht, Netherlands, 1992. · Zbl 0768.58044 [19] J. Kollár, Sharp effective Nullstellensatz , J. Amer. Math. Soc. 1 (1988), no. 4, 963-975. JSTOR: · Zbl 0682.14001 · doi:10.2307/1990996 · links.jstor.org [20] E. Korkina, Classification of $$A$$-graded algebras with $$3$$ generators , Indag. Math. (N.S.) 3 (1992), no. 1, 27-40. · Zbl 0756.13011 · doi:10.1016/0019-3577(92)90025-G [21] E. Korkina, G. Post, and M. Roelofs, Algèbres graduées de type $$A$$ , C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 9, 653-655. · Zbl 0761.13002 [22] S. Lang, Algebra , Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. · Zbl 0193.34701 [23] E. Mayr and A. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals , Adv. in Math. 46 (1982), no. 3, 305-329. · Zbl 0506.03007 · doi:10.1016/0001-8708(82)90048-2 [24] L. Robbiano and M. Sweedler, Subalgebra bases , Commutative algebra (Salvador, 1988), Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 61-87. · Zbl 0725.13013 · doi:10.1007/BFb0085537 [25] R. Stanley, Generalized $$H$$-vectors, intersection cohomology of toric varieties, and related results , Commutative algebra and combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 187-213. · Zbl 0652.52007 [26] B. Sturmfels, Gröbner bases of toric varieties , Tohoku Math. J. (2) 43 (1991), no. 2, 249-261. · Zbl 0714.14034 · doi:10.2748/tmj/1178227496 [27] B. Sturmfels, Asymptotic analysis of toric ideals , Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), no. 2, 217-228. · Zbl 0784.14024 · doi:10.2206/kyushumfs.46.217 [28] R. Thomas, A geometric Buchberger algorithm for integer programming , to appear in Math. Oper. Res. JSTOR: · Zbl 0846.90079 · doi:10.1287/moor.20.4.864 · links.jstor.org [29] S. Xambó, On projective varieties of minimal degree , Collect. Math. 32 (1981), no. 2, 149-163. · Zbl 0501.14020 · www.collectanea.ub.edu · eudml:41150 [30] C. K. Yap, A new lower bound construction for commutative Thue systems with applications , J. Symbolic Comput. 12 (1991), no. 1, 1-27. · Zbl 0731.68060 · doi:10.1016/S0747-7171(08)80138-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.