2020-09-24T06:21:50Zhttps://uvadoc.uva.es/oai/requestoai:uvadoc.uva.es:10324/408082020-05-15T10:41:16Zcom_10324_32197com_10324_952com_10324_894col_10324_32200
00925njm 22002777a 4500
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Matusevich, Laura Felicia
author
Ojeda, Ignacio
author
2018
A congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn 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 Nn 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 Sánchez (J Symbolic Comput 30(4):383–400, 2000).
Greuel, Gert-Martin; Narváez Macarro, Luis; Xambó-Descamps, Sebastià (coords.). Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics: Festschrift for Antonio Campillo on the Occasion of his 65th Birthday. Springer, 2018, p. 429-454
978-3-319-96827-8
http://uvadoc.uva.es/handle/10324/40808
Binomial Ideals and Congruences on Nn