Decoupling for smooth surfaces in ℝ3

abstract: For each $d\geq 0$, we prove decoupling inequalities in $\mathbb{R}^3$ for the graphs of all bivariate polynomials of degree at most $d$ with bounded coefficients, with the decoupling constant uniform in the coefficients of those polynomials. As a consequence, we prove a decoupling inequality for (a compact piece of) every smooth surface in $\mathbb{R}^3$, which in particular solves a conjecture of Bourgain, Demeter, and Kemp.