A generalization of Vinogradov's mean value theorem

Asymptotic estimates for rational spaces on hypersurfaces in function fields

“…Thus the system (1.7) has the same shape as the system (1.3). By applying the Linnik-Karatsuba method and the repeated efficient differencing process, we may obtain results that are of the same strength as the integer analogue considered in [15]. The case when k > p is much more complicated.…”

“…It is therefore natural to ask about linear spaces of higher dimension. Asymptotic estimates for the number of such spaces up to a given height have been considered in recent work of Parsell (see [13], [14], [15], and [16]). Let V be a rational linear space of dimension d Using the multinomial theorem, for each j with 1 ≤ j ≤ s, we have…”

“…Under a similar solubility condition as in [15], we employ a variant of the Hardy-Littlewood circle method to prove the following theorem.…”

“…We shall frequently abbreviate a monomial of the shape x [15], to estimate N s,k,d (P ), we consider a generalization of Vinogradov-type mean value theorem.…”

“…In [15], Parsell applied the Hardy-Littlewood circle method to estimate M s,k,d (P ). In particular, he proved a generalization of Vinogradov's mean value theorem, which concerns the number of solutions of an auxiliary symmetric system For P ∈ R with P > 0, let N s,k (P ) denote the number of solutions of (1.5) in F q [t] s with deg z j < P (1 ≤ j ≤ s).…”