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VINOGRADOV SYSTEMS WITH A SLICE OFF
Author(s) -
Brandes Julia,
Wooley Trevor D.
Publication year - 2017
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579317000134
Subject(s) - mathematics , diagonal , value (mathematics) , pure mathematics , statistics , geometry
LetI s , k , r( X )denote the number of integral solutions of the modified Vinogradov system of equationsx 1 j + ⋯ + x s j = y 1 j + ⋯ + y s j( 1 ⩽ j ⩽ k , j ≠ r ) ,with 1 ⩽ x i , y i ⩽ X( 1 ⩽ i ⩽ s ) . By exploiting sharp estimates for an auxiliary mean value, we obtain bounds forI s , k , r( X )for 1 ⩽ r ⩽ k − 1 . In particular, when s , k ∈ N satisfy k ⩾ 3 and 1 ⩽ s ⩽ ( k 2 − 1 ) / 2 , we establish the essentially diagonal behaviourI s , k , 1( X ) ≪ X s + ɛ.