Restriction estimates for quadratic manifolds of arbitrary codimensions

The restriction conjecture is one of the famous problems in harmonic analysis. There have been many methods developed in the study of the conjecture for the paraboloid. In this paper, we generalize the multilinear method of Bourgain and Guth for the paraboloid, and obtain restriction estimates for all quadratic manifolds of arbitrary codimensions. In particular, our theorem recovers the main theorem of Bourgain and Guth for the paraboloid. A new ingredient is a covering lemma for varieties whose proof relies on Tarski's projection theorem in real algebraic geometry. We also provide algorithms to compute several algebraic quantities that naturally appear in the argument. These algorithms rely on a cylindrical decomposition in real algebraic geometry.