THE DEGREE AND REGULARITY OF VANISHING IDEALS OF ALGEBRAIC TORIC SETS OVER FINITE FIELDS

Let X* be a subset of an affine space ?(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 Ps-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.