Let f: \{0,1\}^n \to \{0, 1\} be a boolean function, and let f_\land (x, y) = f(x \land y) denote the AND-function of f, where x \land y denotes bit-wise AND. We study the deterministic communication complexity of f_\land and show that, up to a \log n factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of f_\land. This comes within a \log n factor of establishing the log-rank conjecture for AND-functions with no assumptions on f. Our result stands in contrast with previous results on special cases of the log-rank
conjecture, which needed significant restrictions on f such as monotonicity or low \mathbb{F}_2-degree. Our techniques can also be used to prove (within a \log n factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of f_\land is polynomially-related to the AND-decision tree complexity of f.

The results rely on a new structural result regarding boolean functions f:\{0, 1\}^n \to \{0, 1\} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing f has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials f:\{0,1\}^n \to \mathbb{R} with a larger range.

Fixed author order.

Let f: \{0,1\}^n \to \{0, 1\} be a boolean function, and let f_\land (x, y) = f(x \land y) denote the AND-function of f, where x \land y denotes bit-wise AND. We study the deterministic communication complexity of f_\land and show that, up to a \log n factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of f_\land. This comes within a \log n factor of establishing the log-rank conjecture for AND-functions with no assumptions on f. Our result stands in contrast with previous results on special cases of the log-rank
conjecture, which needed significant restrictions on f such as monotonicity or low \mathbb{F}_2-degree. Our techniques can also be used to prove (within a \log n factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of f_\land is polynomially-related to the AND-decision tree complexity of f.

The results rely on a new structural result regarding boolean functions f:\{0, 1\}^n \to \{0, 1\} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing f has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials f:\{0,1\}^n \to \mathbb{R} with a larger range.