Hilbert’s seventeenth problem famously showed that not every nonnegative polynomial admits a sum-of-squares (SoS) representation. Hilbert also identified several special classes for which nonnegativity and SoS are equivalent, including univariate polynomials, quadratic polynomials, and bivariate quartics. In this talk, we will extend this connection to new classes of multivariate quartically regularized polynomials arising as subproblems …