# Some Bounds Arising From a Polynomial Ideal Associated to Any t-Design

### Abstract

We consider ordered pairs (X,B) where X is a finite set of size v and B is some collection of k-element subsets of X such that every t-element subset of X is contained in exactly l ``blocks¢¢ B Î B for some fixed l. We represent each block B by a zero-one vector c_{B} of length v and explore the ideal I(B) of polynomials in v variables with complex coefficients which vanish on the set { c_{B} | B Î B}. After setting up the basic theory, we investigate two parameters related to this ideal: g_{1}(B) is the smallest degree of a non-trivial polynomial in the ideal I(B) and g_{2}(B) is the smallest integer s such that I(B) is generated by a set of polynomials of degree at most s. We first prove the general bounds t/2 < g_{1}(B) £ g_{2}(B) £ k. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have g_{2}(B) £ t. But we expect g_{2}(B) to be closer to k for less structured designs and we indicate this by constructing infinitely many triple systems satisfying g_{2}(B)=k.

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