Computer Science and Discrete Mathematics (CSDM)
Let $f(x_1,...,x_n)$ be a low degree polynomial over $F_p$. I will prove that there always exists a small set $S$ of variables, such that `most` Fourier coefficients of $f$ contain some variable from the set $S$. As an application, we will get a derandomized sampling of elements in $F_p^n$ which `look uniform` to $f$.
The talk will be self contained, even though in spirit it is a continuation of my previous talk on pseudorandom generators for $CC0[p]$. Based on joint work with Amir Shpilka and Partha Mukhopadhyay.
In our work we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results.
A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that for $m\geq n$, the rank of any $(q,k,t)$-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least $n - (qtn/2k)^2$ .
Using this result we derive the following applications:
In a sequence of recent papers, Sudan and coauthors have investigated the relation between testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Linear-invariance is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties.
Finding the longest increasing subsequence (LIS) is a classic algorithmic problem. Simple $O(n log n)$ algorithms, based on dynamic programming, are known for solving this problem exactly on arrays of length $n$.
I will talk about the computational complexity of computing the noncommutative determinant. In contrast to the case of commutative algebras, we know of (virtually) no efficient algorithms to compute the determinant over non-commutative domains. Our results show that the determinant in noncommutative settings can be as hard as the permanent.
I will describe the proof of the following surprising result: the typical billiard paths form the family of the most uniformly distributed curves in the unit square. I will justify this vague claim with a precise statement. As a byproduct, we obtain the counter-intuitive fact that the complexity of the test set is almost irrelevant. The error term is shockingly small, and it does not matter that we test uniformity with a nice set (like a circle or a square), or with an arbitrarily ugly Lebesgue measurable subset of the unit square.