text: We consider ordered pairs @math where @math is a finite set of size @math and @math is some collection of @math -element subsets of @math such that every @math -element subset of @math is contained in exactly @math "blocks" @math for some fixed @math . We represent each block @math by a zero-one vector @math of length @math and explore the ideal @math of polynomials in @math variables with complex coefficients which vanish on the set @math . After setting up the basic theory, we investigate two parameters related to this ideal: @math is the smallest degree of a non-trivial polynomial in the ideal @math and @math is the smallest integer @math such that @math is generated by a set of polynomials of degree at most @math . We first prove the general bounds @math . Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have @math . But we expect @math to be closer to @math for less structured designs and we indicate this by constructing infinitely many triple systems satisfying @math .
  @cite_5: Arithmetic combinatorics is often concerned with the problem of controlling the possible range of behaviours of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of controlling the possible range of behaviours of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in nature. In recent years, however, many outstanding problems in these directions have been solved by algebraic means (and more specifically, using tools from algebraic geometry and or algebraic topology), giving rise to an emerging set of techniques which is now known as the polynomial method. Broadly speaking, the strategy is to capture (or at least partition) the arbitrary sets of objects (viewed as points in some configuration space) in the zero set of a polynomial whose degree (or other measure of complexity) is under control; for instance, the degree may be bounded by some function of the number of objects. One then uses tools from algebraic geometry to understand the structure of this zero set, and thence to control the original sets of objects. While various instances of the polynomial method have been known for decades (e.g. Stepanov’s method, the combinatorial Nullstellensatz, or Baker’s theorem), the general theory of this method is still in the process of maturing; in particular, the limitations of the polynomial method are not well understood, and there is still considerable scope to apply deeper results from algebraic geometry or algebraic topology to strengthen the method further. In this survey we present several of the known applications of these methods, focusing on the simplest cases to illustrate the techniques. We will assume as little prior knowledge of algebraic geometry as possible.
  @cite_10: We show that for integer @math , any subset @math free of three-term arithmetic progressions has size @math , with an absolute constant @math .
  @cite_12: 
summary: The polynomial method'' in combinatorics @cite_5 is a powerful tool for deriving bounds on the size of combinatorial objects and for proving non-existence of extremal objects. In particular, a recent breakthrough by Croot, Lev and Pach @cite_10 (see also @cite_12 ) has stimulated interest in collections of multivariate polynomials that vanish on some configuration in a finite vector space. Here, we work in characteristic zero and are interested in how the ideal we obtain is related to the structure of the design. By contrast, the authors of @cite_10 @cite_12 work over fields of positive characteristic. To see the connection, we note that, since our zero set lies in @math , the polynomial generators @math may always be chosen from @math . So application of the reduction @math maps the ideal @math into the ideal of the same finite variety considered modulo @math . It will be an interesting follow-up task to determine when the image under this map is the full ideal in positive characteristic.