Binomial Ideals and Congruences on $$mathbb {N}^n$$ Chapter uri icon

abstract

  • Springer Nature Switzerland AG 2018. All Rights Reserved. congruence on n is an equivalence relation on n that is compatible with the additive structure. If K is a field, and I is a binomial ideal in K[X 1 , . . ., X n ] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on n by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of X u and X v that belongs to I . While every congruence on n arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on n are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297-1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1-45, 1996) and Ojeda and Piedra Snchez (J Symbolic Comput 30(4):383-400, 2000).

author list (cited authors)

  • Matusevich, L. F., & Ojeda, I.

complete list of authors

  • Matusevich, Laura Felicia||Ojeda, Ignacio

Book Title

  • Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

publication date

  • September 2018