Vinogradov Systems With a Slice Off

Brandes, Julia; Wooley, Trevor D.

doi:10.1112/s0025579317000134

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…As mentioned in the introduction, a close reading of our methods reveals that the translation-invariant structure of the system does indeed play a crucial role in our arguments. Consequently, there is little hope to extend our results to incomplete Vinogradov systems such as those considered in [3] without fundamentally different ideas.…”

Section: Further Discussion: Generalisations and Limitationsmentioning

confidence: 99%

On the Inhomogeneous Vinogradov System

Brandes

Hughes²

2022

Bull. Aust. Math. Soc.

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We show that the system of equations $$ \begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*} $$ has appreciably fewer solutions in the subcritical range $s < \tfrac 12k(k+1)$ than its homogeneous counterpart, provided that $a_{\ell } \neq 0$ for some $\ell \leqslant k-1$ . Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.

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Section: Further Discussion: Generalisations and Limitationsmentioning

confidence: 99%

On the Inhomogeneous Vinogradov System

Brandes

Hughes²

2022

Bull. Aust. Math. Soc.

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“…Observe that Theorem 1.1 is a special case of Theorem 1.6. The proofs of our results rest on an idea that played a crucial role in the second author's work on pairs of quadratic and cubic diagonal equations [12], and which has been explored further in the authors' recent work on incomplete Vinogradov systems [4]. In these papers, the missing linear equation is artificially added in, which makes it possible to exploit the strong bounds on Vinogradov's mean value theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Our second special case concerns systems of higher degree k. In a recent paper [4], we studied Vinogradov systems lacking the linear slice and established diagonal behaviour for the mean value I s,k,1 (X) for s (k 2 − 1)/2, thus missing the critical point s = k(k + 1)/2 − 1 only by a term linear in k. It turns out, however, that under suitable congruence conditions the full main conjecture is true when one adds in one additional quadratic equation.…”

Section: Introductionmentioning

confidence: 99%

Optimal mean value estimates beyond Vinogradov's mean value theorem

Brandes¹,

Wooley²

2019

Preprint

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We establish sharp mean value estimates associated with the number of integer solutions of certain systems of diagonal equations. This is the first occasion on which bounds of this quality have been attained for Diophantine systems not of Vinogradov type. As a consequence of this progress, whenever u 3v we obtain the Hasse principle for systems consisting of v cubic and u quadratic diagonal equations in 6v + 4u + 1 variables, thus attaining the convexity barrier for this problem.

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“…The proofs of our results rest on an idea that played a crucial role in the second author's work on pairs of quadratic and cubic diagonal equations [12], and which has been explored further in the authors' recent work on incomplete Vinogradov systems [4]. In these papers, the missing linear equation is artificially added in, which makes it possible to exploit the strong bounds on Vinogradov's mean value theorem.…”

mentioning

confidence: 99%