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Given an integer $q$ and a polynomial $f\in \mathbb{Z}_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb{Z}_q=\mathbb{Z}/q\mathbb{Z},$ we obtain new results concerning the number of solutions to congruences of the form $y\equiv f(x) \Mod{q},$ with integer variables lying in some cube $\mathcal{B}$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently, using transference principles from the geometry of numbers. We also use a variant of the main conjecture for the Vinogradov mean value theorem of Bourgain, Demeter and Guth and of Wooley, which allows one to deal with solutions to the Vinogradov mean value theorem when the variables run through rather sparse sets.
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