Let $V$ be any vector space of multivariate degree-$d$ homogeneous\npolynomials with co-dimension at most $k$, and $S$ be the set of points where\nall polynomials in $V$ {\\em nearly} vanish. We establish a qualitatively\noptimal upper bound on the size of $\\epsilon$-covers for $S$, in the\n$\\ell_2$-norm. Roughly speaking, we show that there exists an $\\epsilon$-cover\nfor $S$ of cardinality $M = (k/\\epsilon)^{O_d(k^{1/d})}$. Our result is\nconstructive yielding an algorithm to compute such an $\\epsilon$-cover that\nruns in time $\\mathrm{poly}(M)$.\n Building on our structural result, we obtain significantly improved learning\nalgorithms for several fundamental high-dimensional probabilistic models with\nhidden variables. These include density and parameter estimation for\n$k$-mixtures of spherical Gaussians (with known common covariance), PAC\nlearning one-hidden-layer ReLU networks with $k$ hidden units (under the\nGaussian distribution), density and parameter estimation for $k$-mixtures of\nlinear regressions (with Gaussian covariates), and parameter estimation for\n$k$-mixtures of hyperplanes. Our algorithms run in time {\\em quasi-polynomial}\nin the parameter $k$. Previous algorithms for these problems had running times\nexponential in $k^{\\Omega(1)}$.\n At a high-level our algorithms for all these learning problems work as\nfollows: By computing the low-degree moments of the hidden parameters, we are\nable to find a vector space of polynomials that nearly vanish on the unknown\nparameters. Our structural result allows us to compute a quasi-polynomial sized\ncover for the set of hidden parameters, which we exploit in our learning\nalgorithms.\n