  Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We
investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1
- \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of
variables needed to ensure that if $\eta$ is real and $\tau > 0$ is
sufficiently large then there exist integers $x_1 > \mu_1, \ldots, x_s > \mu_s$
such that $|\mathfrak{F}(\mathbf{x}) - \tau| < \eta$. This is a real analogue
to Waring's problem. When $s \ge 2k^2-2k+3$, we provide an asymptotic formula.
We prove similar results for sums of general univariate degree $k$ polynomials.
