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

TL;DR

This paper advances the understanding of extension estimates for paraboloids over finite fields by establishing new bounds and extending previous results to higher dimensions, thereby clarifying key conjectures in the field.

Contribution

It introduces improved extension theorems for paraboloids in finite fields, extending prior results to higher dimensions and clarifying related conjectures.

Findings

Improved L^2 to L^r extension bounds for paraboloids in higher dimensions.

Extension of 3-D paraboloid results to higher dimensions.

Clarification of conjectures on finite field extension problems.

Abstract

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