  We study the complete intersection property and the algebraic invariants
(index of regularity, degree) of vanishing ideals on degenerate tori over
finite fields. We establish a correspondence between vanishing ideals and toric
ideals associated to numerical semigroups. This correspondence is shown to
preserve the complete intersection property, and allows us to use some
available algorithms to determine whether a given vanishing ideal is a complete
intersection. We give formulae for the degree, and for the index of regularity
of a complete intersection in terms of the Frobenius number and the generators
of a numerical semigroup.
