On the inhomogeneous Vinogradov system

Brandes, Julia; Hughes, Kevin

Abstract

We show that the system of equations

∑_{i=1}^{s} (x_i^j−y_i^j) = a_j (1⩽j⩽k)

has appreciably fewer solutions in the subcritical range s<k(k+1)/2
than its homogeneous counterpart, provided that a_ℓ≠0 for some ℓ⩽k−1. Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.

Citation

Brandes, J., & Hughes, K. (2022). On the inhomogeneous Vinogradov system. Bulletin of the Australian Mathematical Society, 106(3), 396-403. https://doi.org/10.1017/s0004972722000284

• Journal Article Type: Article
• Acceptance Date: Feb 15, 2022
• Online Publication Date: Apr 19, 2022
• Publication Date: 2022-12
• Deposit Date: Dec 2, 2022
• Publicly Available Date: Dec 2, 2022
• Journal: Bulletin of the Australian Mathematical Society
• Print ISSN: 0004-9727
• Electronic ISSN: 1755-1633
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 106
• Issue: 3
• Pages: 396-403
• DOI: https://doi.org/10.1017/s0004972722000284
• Keywords: Diophantine equations, exponential sums
• Public URL: http://researchrepository.napier.ac.uk/Output/2965313