Theory of Computing ------------------- Title : Dual Polynomials for Collision and Element Distinctness Authors : Mark Bun and Justin Thaler Volume : 12 Number : 16 Pages : 1-34 URL : https://theoryofcomputing.org/articles/v012a016 Abstract -------- The approximate degree of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ is the minimum degree of a real polynomial that approximates $f$ to within error $1/3$ in the $\ell_\infty$ norm. In an influential result, Aaronson and Shi (J. ACM, 2004) proved tight $\tilde{\Omega}(n^{1/3})$ and $\tilde{\Omega}(n^{2/3})$ lower bounds on the approximate degree of the Collision and ElementDistinctness functions, respectively. Their proof was non- constructive, using a sophisticated symmetrization argument and tools from approximation theory. More recently, several open problems in the study of approximate degree have been resolved via the construction of dual polynomials. These are explicit dual solutions to an appropriate linear program that captures the approximate degree of any function. We reprove Aaronson and Shi's results by constructing explicit dual polynomials for the Collision and ElementDistinctness functions. Our constructions are heavily inspired by Kutin's (Theory of Computing, 2005) refinement and simplification of Aaronson and Shi's results.