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Regularity and algebraic properties of certain lattice ideals. (English) Zbl 1309.13027

Summary: We study the regularity and the algebraic properties of certain lattice ideals. We establish a map \(I \mapsto \tilde I\) between the family of graded lattice ideals in an \(\mathbb{N}\)-graded polynomial ring over a field \(K\) and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of \(I\) but not the degree. We relate the Hilbert series and the generators of \(I\) and \(\tilde I\). If dim(\(I\)) = 1, we relate the degrees of \(I\) and \(\tilde I\). It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields,with prescribed regularity and degree of a certain type. Let \(X\) be a subset of a projective space over a field \(K\). It is shown that the vanishing ideal of \(X\) is a lattice ideal of dimension 1 if and only if \(X\) is a finite subgroup of a projective torus. For finite fields, it is shown that \(X\) is a subgroup of a projective torus if and only if \(X\) is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartite graph in terms of the regularities of the vanishing ideals of the blocks of the graph.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
14H45 Special algebraic curves and curves of low genus
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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