The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Abstract

Let X* be a subset of an affine space 𝔸 ^s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x → [x] and x → [(x, 1)], respectively, where [x] and [(x, 1)] are points in the projective spaces ℙ ^s−1 and ℙ ^s , respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud–Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.

Key Words:

• Bipartite graph
• Clutter
• Complete intersection
• Degree
• Evaluation code
• Minimum distance
• Regularity
• Vanishing ideal

2010 Mathematics Subject Classification:

• Primary 13F20
• Secondary 13P25
• 11T71
• 94B25

ACKNOWLEDGEMENT

The first author is a member of the Center for Mathematical Analysis, Geometry, and Dynamical Systems. The second author was partially supported by SNI.

Notes

Communicated by S. Goto.