Cellular Binomial Ideals. Primary Decomposition of Binomial Ideals

It is known that algorithms exist which compute primary decompositions of polynomial ideals (Gianni et al., 1988; Eisenbud et al., 1992; Becker and Weispfenning, 1993; and more recently Shimoyama and Yokoyama, 1996). However, in case the ideal is binomial, binomiality of its primary components is not assured, that is, the above algorithms do not necessarily compute a decomposition into binomial components even if such a decomposition exists. The older algorithms immediately leave the category of binomial ideals. For example, Algorithm ZPDF in Gianni et al. (1988) and NORMPOS in Becker and Weispfenning (1993) make changes of coordinates and the algorithms in Eisenbud et al. (1992) use syzygy computations and Jacobian ideals. On the other hand, the binomial ideal (xy − z, xy − z, x − yz) has a binomial primary decomposition in Q[x, y, z] (see Example B.2), but the algorithm by Shimoyama and Yokoyama (1996) (implemented in Singular Greuel et al., 1998) does not yield a binomial primary decomposition. Eisenbud and Sturmfels (1996) show that, over an algebraically closed field, binomial primary decompositions of binomial ideals exist. However, these authors do not complete their algorithms in several steps in which it is necessary to know a sufficiently large integer which verifies certain properties, thus giving rise to some theoretical problems. In this paper we give a solution to these problems and we fill all the gaps in the algorithms in Eisenbud and Sturmfels (1996). In Appendix A, we present the algorithms for decomposing binomial ideals that emerge from the general theory. We start with a binomial ideal I in S = k[x1, . . . , xn] where k is an algebraically closed