The threshold degree of a Boolean function
$f\colon\{0,1\}\to\{-1,+1\}$ is the least degree of a real
polynomial $p$ such $f(x)\equiv\mathrm{sgn}\; p(x).$ We
construct two halfspaces on $\{0,1\}^n$ whose intersection has
threshold degree $\Theta(\sqrt n),$ an exponential improvement on
previous lower bounds. This solves an open problem due to Klivans
(2002) and ... more >>>

We study the approximation of halfspaces $h:\{0,1\}^n\to\{0,1\}$ in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the "hardest" halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all ... more >>>