Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is NP$^{\#\rm P}$ (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). ... more >>>

Given a Boolean function $f: \{-1,1\}^n\rightarrow \{-1,1\}$, define the Fourier distribution to be the distribution on subsets of $[n]$, where each $S\subseteq [n]$ is sampled with probability $\widehat{f}(S)^2$. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] seeks to relate two fundamental measures associated with the Fourier distribution: does ... more >>>