We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions $f$, it is conjectured that $\mathrm{rdeg}(f)$ is polynomially related to $\mathrm{deg}(f)$, where $\mathrm{deg}(f)$ is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least $\mathrm{deg}(f)/2$ and ... more >>>

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function $f:\{\pm1\}^{n} \to \{\pm1\}$ we give a poly$(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from ... more >>>