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 .
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.
@cite_5 @cite_10 @cite_12
2084050956 1976392590 2016576580 2104223485
We show that, over Q, if an n-variate polynomial of degree d = nO(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O(√(d log n log d log s))) (respectively of size exp(O(√(d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O(√(d log d))) computing the d × d determinant Detd. It also means that if we can prove a lower bound of exp(omega(√(d log d))) on the size of any ΣΠΣ-circuit computing the d × d permanent Permd then we get super polynomial lower bounds for the size of any arithmetic branching program computing Permd. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Detd or Perm_d, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)). Multipolynomial resultants provide the most efficient methods known (in terms as asymptoticcomplexity) for solving certain systems of polynomial equations or eliminating variables (, 1988). The resultant of f"1, ..., f"n in K[x"1,...,x"m] will be a polynomial in m-n+1 variables which is zero when the system f"1=0 has a solution in ^m ( the algebraic closure of K). Thus the resultant defines a projection operator from ^m to ^(^m^-^n^+^1^). However, resultants are only exact conditions for homogeneous systems, and in the affine case just mentioned, the resultant may be zero even if the system has no affine solution. This is most serious when the solution set of the system of polynomials has ''excess components'' (components of dimension >m-n), which may not even be affine, since these cause the resultant to vanish identically. In this paper we describe a projection operator which is not identically zero, but which is guaranteed to vanish on all the proper (dimension=m-n) components of the system f"i=0. Thus it fills the role of a general affine projection operator or variable elimination ''black box'' which can be used for arbitrary polynomial systems. The construction is based on a generalisation of the characteristic polynomial of a linear system to polynomial systems. As a corollary, we give a single-exponential time method for finding all the isolated solution points of a system of polynomials, even in the presence of infinitely many solutions, at infinity or elsewhere. In their paper on the ''chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o^(^m^) also admit arithmetic circuits of depth four and size 2^o^(^m^). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of nxn matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O^(^n^l^o^g^n^). If the original circuit uses only integer constants of polynomial size, then the same is true for the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: -the existence of nontrivial boolean circuits of constant depth for languages in LOGCFL; -reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree. Let f 1 , ldots, f s be polynomials in Q[X 1 , ..., X n ] that generate a radical ideal and let V be their complex zero-set. Suppose that V is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of V gives at least one point in each connected component of V ∩ Rn. We deduce an algorithm that extends that of Bank, Giusti, Heintz and Mbakop to non-compact situations. Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity.