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On Generators of Ideals Associated with Unions of Linear Varieties

Consider the polynomial ring /?[*,, ...,xn] over a unique factorization domain R. A form (i.e., a homogeneous polynomial) is said to split if it is a product of linear forms. When a homogeneous ideal is generated by splitting forms, the associated projective algebraic set is a finite union of linear subvarieties of P"~(R). But conversely, when a projective algebraic set decomposes into linear subvarieties, its associated radical ideal may not be generated by splitting forms. In this paper we construct a recursive algorithm for establishing sufficient conditions for an ideal to be generated by a prescribed set of splitting forms and apply this algorithm to a family of ideals that have arisen in the study of block designs. Our results on ideal generators have very interesting applications to graph theory, which are discussed elsewhere.