https://uvadoc.uva.es/handle/10324/32200 IMUVA - Capítulos de Monografías 2026-04-17T11:47:06Z https://uvadoc.uva.es/handle/10324/40808 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). 2018-01-01T00:00:00Z https://uvadoc.uva.es/handle/10324/36079 We introduce the Poincar´e polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. We prove that the Poincar´e polynomial is a complete invariant of the code in the binary and ternary case and it is not when q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for binary Reed-Muller codes. 2018-01-01T00:00:00Z https://uvadoc.uva.es/handle/10324/35953 In this note we announce a result determining the Newton–Okounkov bodies of the line bundle OP2 (1) with respect to exceptional curve plane valuations non-positive at infinity 2018-01-01T00:00:00Z