On the inhomogeneous Vinogradov system

2021/10/05 by Brandes, Julia, Hughes, Kevin · 1 citation
#11L15 #11P05 #42B05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11D45. Secondary: 11D85

paper · doi:10.48550/arxiv.2110.02366

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.

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