On Generators of Ideals Associated with Unions of Linear Varieties

Consider the polynomial ring Rx₁, ..., x_n] over a unique factorizationdomain R. A form (i.e., a homogeneous polynomial) is said tosplit if it is a product of linear forms. When a homogeneousideal is generated by splitting forms, the associated projectivealgebraic set is a finite union of linear subvarieties of P^n–1(R).But conversely, when a projective algebraic set decomposes intolinear subvarieties, its associated radical ideal may not begenerated by splitting forms. In this paper we construct a recursivealgorithm for establishing sufficient conditions for an idealto be generated by a prescribed set of splitting forms and applythis algorithm to a family of ideals that have arisen in thestudy of block designs. Our results on ideal generators havevery interesting applications to graph theory, which are discussedelsewhere.