A commutative algebraic approach to the fitting problem

Given a finite set of points $\Gamma$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(\Gamma)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s). If $\Gamma$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(\Gamma)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $\Gamma$. Note that in $\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given set of points $\Gamma\subset\mathbb P^2$ equals the index of nilpotency of the corresponding ideal, plus one.Comment: 8 page

Cayley-Bacharach and evaluation codes on complete intersections

In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the projective plane. In this paper, we generalize Hansen's results from P^2 to P^m; we also show that the hypotheses in Hansen's work may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach theorem and bounds on evaluation codes obtained from reduced zero-schemes.Comment: 10 pages. v2: minor expository change

Complete intersections in binomial and lattice ideals

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.Comment: Internat. J. Algebra Comput., to appea

Complete intersection vanishing ideals on degenerate tori over finite fields

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.Comment: Arabian Journal of Mathematics, to appea

Computing the degree of a lattice ideal of dimension one

We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.Comment: J. Symbolic Comput., to appea

Bounding invariants of fat points using a coding theory construction

Let Z \subseteq \proj{n} be a fat points scheme, and let $d(Z)$ be the minimum distance of the linear code constructed from $Z$. We show that $d(Z)$ imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal free resolution of $I_Z$, the defining ideal of $Z$. We investigate this relation in the case that the support of $Z$ is a complete intersection; when $Z$ is reduced and a complete intersection we give lower bounds for $d(Z)$ that improve upon known bounds.Comment: 18 pages, 1 figure; accepted in J. Pure Appl. Algebr

Regularity and algebraic properties of certain lattice ideals

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an 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 I\~. If dim(I)=1, we relate the degrees of I and 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 bipartie graph in terms of the regularities of the vanishing ideals of the blocks of the graph.Comment: Bull. Braz. Math. Soc. (N.S.), to appea