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We show that when $\\varphi_j\\in \\mathbb Z[t]$ $(1\\le j\\le k)$ is a system of polynomials with non-vanishing Wronskian, and $s\\le k(k+1)/2$, then for all complex sequences $(\\mathfrak a_n)$, and for each $\\epsilon>0$, one has \\[ \\int_{[0,1)^k} \\left| \\sum_{|n|\\le X} {\\mathfrak a}_n e(\\alpha_1\\varphi_1(n)+\\ldots +\\alpha_k\\varphi_k(n)) \\right|^{2s} {\\rm d}{\\boldsymbol \\alpha} \\ll X^\\epsilon \\left( \\sum_{|n|\\le X} |{\\mathfrak a}_n|^2\\right)^s. \\] As a special case of this result, w","authors_text":"Trevor D. 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