On Functions of Strength

Define a subset S ~ P(X) to be independent if and only if {z~[BC S} is a linearly independent subset of V. This clearly induces a matroid structure on P(X). This matroid has been studied in connection with many problems in the theory of designs or extremal problems for families of finite sets. In fact many interesting inequalities are proved by showing that certain families of subsets of X correspond to independent sets in this matroid (see [1, 3, 4, 5, 6, 7, 12]). Tlae aim of this paper is to study circuits in this matroid. In this section we first redefine the matroid described above in 'dual' terms which seem more suitable and familiar and then describe various results. The next section is devoted to the proofs. For any subset Ac=X, define a monomial p (A)= 11 .v, P ( 0 ) = I . We xEA identify A with p(A) and in this way we also think of P(X) as the set of all squarefree monomials in xl, x2 . . . . . x,. Similarly we also consider any fE V(X) as a polynomial f = ~ ' f(A)p(A). Thus V(X) is also free module of all square-free polynoAC=X