Sharp endpoint extension inequalities for the moment curve on finite fields

TL;DR

This paper establishes the best possible bounds for endpoint extension inequalities related to the moment curve over finite fields, identifying optimal constants and maximizers in specific regimes.

Contribution

It determines the sharp endpoint extension inequality constants and characterizes maximizers for the moment curve in finite fields, covering low dimensions and large field sizes.

Findings

Optimal constants for the inequality are identified.

Maximizers are characterized in two regimes.

The proof combines analysis, algebra, and combinatorics.

Abstract

We investigate the sharp endpoint extension inequality for the moment curve in finite fields. We determine the optimal constant and characterize the maximizers in two complementary regimes: (i) low dimensions $d\leq 20$; (ii) large field cardinality $q\geq \frac{d(d-1)}{2 \log 6} + \frac{(2d-1)}{3}$. Our proof strategy relies on an intriguing interplay between analysis, algebra and combinatorics.