We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider
f
(
x
1
,
…
,
x
N
)
f(x_1, \dots , x_N)
, where
x
i
∈
R
d
x_i \in \mathbb {R}^d
, and
f
f
is invariant under permutations of its
N
N
arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function
f
f
, and in particular study the dependence of that ratio on
d
,
N
d, N
and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where
N
N
becomes a parameter of the input.