Abstract
Given a family \(\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d\) of d distinct nonconstant polynomials, a positive integer \(k\le d\) and a real positive parameter \(\rho\), we consider the mean value
of exponential sums
where \({\rm x} = (x_1, \ldots, x_k)\) and \({\rm y} =(y_1, \ldots, y_{d-k})\). The case of polynomials \(\varphi_i(T) = T^i, i =1, \ldots, d\) and \(k=d\) corresponds to the classical Vinaogradov mean value theorem.
Here motivated by recent works of Wooley [14] and the authors [9] on bounds on \({\rm sup}_{{\rm y} \in [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; N) |\) for almost all \({\rm x} \in [0,1]^k\), we obtain nontrivial bounds on \(M_{k, \rho} (\varphi, N)\).
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Acknowledgments
The authors are grateful to Angel Kumchev for useful discussions and comparison our work with the settings of [1,2].
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This work was supported by ARC Grant DP170100786.
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Chen, C., Shparlinski, I.E. On a hybrid version of the Vinogradov mean value theorem. Acta Math. Hungar. 163, 1–17 (2021). https://doi.org/10.1007/s10474-020-01111-9
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DOI: https://doi.org/10.1007/s10474-020-01111-9