  We consider a translation and dilation invariant system consisting of k
diagonal equations of degrees 1,2,...,k with integer coefficients in s
variables, where s is sufficiently large in terms of k. We show via the
Hardy-Littlewood circle method that if a subset A of the natural numbers
restricted to the interval [1,N] satisfies Gowers' definition of uniformity of
degree k, then it furnishes roughly the expected number of simultaneous
solutions to the given equations. If A furnishes no non-trivial solutions to
the given system, then we show that the number of elements of A in [1,N] grows
no faster than a constant multiple of N/(log log N)^{-c} as N grows to
infinity, where c>0 is a constant dependent only on k. In particular, we show
that the density of A in [1,N] tends to 0 as N tends to infinity.
