We use a combination of the Hardy-Littlewood circle method and the methods developed by Gowers in his recent proof of Szemer´edi’s Theorem on long arithmetic progressions to obtain quantitative estimates for the upper density of a set of integers containing no solutions to a translation and dilation invariant system of diagonal polynomials of degrees 1, 2, . . . , k. We will also explore the question of finding a bound for the upper density of a subset of the primes containing no solutions to a system of this type in the case k = 2.