Binomial Ideals and Congruences on $$\mathbb {N}^n$$

Laura Felicia Matusevich () and Ignacio Ojeda ()
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Laura Felicia Matusevich: Texas A&M University, Mathematics Department
Ignacio Ojeda: Universidad de Extremadura, Departamento de Matemáticas

A chapter in Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 2018, pp 429-454 from Springer

Abstract: Abstract A congruence on ℕ n $$\mathbb {N}^n$$ is an equivalence relation on ℕ n $$\mathbb {N}^n$$ that is compatible with the additive structure. If 𝕜 $$\Bbbk $$ is a field, and I is a binomial ideal in 𝕜 [ X 1 , … , X n ] $$\Bbbk [X_1,\dots ,X_n]$$ (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on ℕ n $$\mathbb {N}^n$$ 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 n $$\mathbb {N}^n$$ 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 n $$\mathbb {N}^n$$ 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 Sánchez (J Symbolic Comput 30(4):383–400, 2000).

Keywords: Binomial Identity; Algebraic Intuition; Ideal Quotient; Prime Congruence; Monoid Identity (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-96827-8_18

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DOI: 10.1007/978-3-319-96827-8_18

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