We examine n×n matrices over Zm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n 1/(p−1) − O(1)). If m is a non-prime-power integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c √ log n log log n). We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs. Keywords: composite modulus, explicit Ramsey-graph constructions, matrices over rings, co-diagonal matrices 1

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if H is a set-system, which satisfies that for some k, the k-wise intersections occupy only ` residue-classes modulo a p prime, while the sizes of the members of H are not in these residue classes, then the size of H is at most (k \Gamma 1) ` X i=0 / n i ! This result considerably strengthens an upper bound of Furedi (1983), and gives partial answer to a question of T. S'os (1976). As an application, we give explicit constructions for coloring the k-subsets of an n element set with t colors, such that no monochromatic complete hypergraph on exp(c k (log n log log n) 1=t ) vertices exists. By our best knowledge, this is the first explicit construction of a Ramsey-hypergraph.

We study set systems satisfying Frankl–Wilson-type conditions modulo prime powers. We prove that the size of such set systems is polynomially bounded, in contrast with V. Grolmusz’s recent result that for non-prime-power moduli, no polynomial bound exists. More precisely we prove the following result. Theorem. Let p be a prime and q = pk. Let µ1,..., µs be distinct integers, 0 ≤ µi ≤ q −1. Let X be a set of n elements and let A1, A2,..., Am be subsets of X with the following properties: • |Ai | ̸ ≡ µℓ (mod q) for all i, ℓ, 1 ≤ i ≤ m, 1 ≤ ℓ ≤ s. • For all i, j (1 ≤ i < j ≤ m), there exists ℓ (1 ≤ ℓ ≤ s) such that Then where D ≤ 2 s−1. m ≤ |Ai ∩ Aj | ≡ µℓ (mod q). Then

Elementary symmetric polynomials S n are used as a benchmark for the boundeddepth arithmetic circuit model of computation. In this work weprovethatS n modulo composite numbers m = p 1 p 2 can be computed with muchfewer multiplications than over any field, if the coefficients of monomials x i 1 x i 2 i k are allowed to be 1 either mod p 1 or mod p 2 but not necessarily both. More exactly,weprove that for any constant k such a representation of S n can be computed modulo p 1 p 2 using only exp(O( p log n log log n)) multiplications on the most restricted depth-3 arithmetic circuits, for min(p 1 #p 2 ) ?k!.

We describe an explicit construction which, for some fixed absolute positive constant ε, produces, for every integer s> 1 and all sufficiently large m, a graph on at least m ε √ log s / log log s vertices containing neither a clique of size s nor an independent set of size m. 1

Let f be an n variable polynomial with positive integer coefficients, and let H = fH 1 ; H 2 ; : : : ; Hm g be a set-system on the n-element universe. We define set-system f(H) = fG 1 ; G 2 ; : : : ; Gm g, and prove that f(H i1 "H i2 ": : :"H ik ) = jG i1 "G i2 ": : :"G ik j, for any 1 k m, where f(H i1 " H i2 " : : : " H ik ) denotes the value of f on the characteristic vector of H i1 " H i2 " : : : " H ik . The construction of f(H) is a straightforward polynomial--time algorithm from H and polynomial f . In this paper we use this algorithm for constructing set-systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some Ray-Chaudhuri--Wilson-like theorems are proved. Keywords: set-systems, algorithmic constructions, multi-variate polynomials, diadic decomposition, matrix-rank 1

The problem of polynomial interpolation is to reconstruct a polynomial based on its valuations on a set of inputs I. We consider the problem over Zm when m is composite. We ask the question: Given I ⊆ Zm, how many evaluations of a polynomial at points in I are required to compute its value at every point in I? Surprisingly for composite m, this number can vary exponentially between log |I| and |I | in contrast to the prime case where |I | evaluations are necessary. While this minimization problem is NP-hard, we give an efficient algorithm of query complexity within a factor t of the optimum where t is the number of prime factors of m. We use our interpolation algorithm to design algorithms for zero-testing and distributional learning of polynomials over Zm. In some cases, we get an exponential improvement over known algorithms in query complexity and running time. Our main technical contribution is the notion of an interpolating set for I which is a subset S of I such that a polynomial which is 0 over S must be 0 at every point in I. Any interpolation algorithm needs to query an interpolating set for I. Our query-efficient algorithms are obtained by constructing interpolating sets whose size is close to optimal.

In a previous work [4] we found a relation between the ranks of codiagonal matrices (matrices with 0's in their diagonal and non-zeroes elsewhere) and the quality of explicit Ramsey-graph constructions.

Abstract. We give an introduction to the area of “randomness extraction” and survey the main concepts of this area: deterministic extractors, seeded extractors and multiple sources extractors. For each one we briefly discuss background, definitions, explicit constructions and applications. 1

Every Boolean function on n variables can be expressed as a unique multivariate polynomial modulo p for every prime p. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree modulo p must have high complexity in every other characteristic q. More precisely, we show the following results about Boolean functions f: {0, 1} n → {0, 1} which depend on all n variables, and distinct primes p, q: • If f has degree o(log n) modulo p, then it must have degree Ω(n 1−o(1) ) modulo q. Thus a Boolean function has degree o(log n) in only one characteristic. This result is essentially tight as there exist functions that have degree log n in every characteristic. • If f has degree d = o(log n) modulo p, it cannot be computed correctly on more than 1 − p−O(d) fraction of the hypercube by polynomials of degree n 1 2 −ɛ modulo