Let $k_i\ (i=1,2,\ldots ,t)$ be natural numbers with $k_1>k_2>\cdots >k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $\alpha _{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape $$\begin{align*} &\varphi_i(x)=\alpha_{1i}x^{k_1}+\alpha_{2i}x^{k_2}+\cdots+\alpha_{ti}x^{k_t},\end{align*}$$and derive upper bounds for fractional parts of polynomials in the shape $$\begin{align*} &\varphi_1(x_1)+\varphi_2(x_2)+\cdots+\varphi_s(x_s),\end{align*}$$by applying novel mean value estimates related to Vinogradov’s mean value theorem. Our results improve on earlier Theorems of Baker (2017).
 Award ID(s):
 2054503
 NSFPAR ID:
 10405082
 Date Published:
 Journal Name:
 The Electronic Journal of Combinatorics
 Volume:
 30
 Issue:
 1
 ISSN:
 10778926
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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