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We study the extension estimates for paraboloids in d-dimensional vectorspaces over finite fields F_q with q elements. We use the connection betweenL^2 based restriction estimates and L^p\to L^r extension estimates forparaboloids. As a consequence, we improve the L^2\to L^r extension resultsobtained by A. Lewko and M. Lewko in even dimensions d\ge 6 and odd dimensionsd=4\ell+3 for \ell \in \mathbb N. Our results extend the consequences for 3-Dparaboloids due to M. Lewko to higher dimensions. We also clarifies conjectureson finite field extension problems for paraboloids.

Conjecture and improved extension theorems for paraboloids in the finite field setting