The local solubility for homogeneous polynomials with random coefficients over thin sets

Abstract

Let $$ and $$ be natural numbers greater or equal to 2. Let $$ be a homogeneous polynomial in $$ variables of degree $$ with integer coefficients $$, where $$ denotes the inner product, and $$ denotes the Veronese embedding with $$. Consider a variety $$ in $$, defined by $$. In this paper, we examine a set of integer vectors $$, defined by