An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let
S
⊆
Z
l
S\subseteq \mathbb {Z}^l
be a set of positive upper Banach density, let
p
1
(
n
)
,
…
,
p
k
(
n
)
p_1(n),\dotsc ,p_k(n)
be polynomials with rational coefficients taking integer values on the integers and satisfying
p
i
(
0
)
=
0
p_i(0)=0
,
i
=
1
,
…
,
k
;
i=1,\dotsc ,k;
then for any
v
1
,
…
,
v
k
∈
Z
l
v_1,\dotsc ,v_k\in \mathbb {Z}^l
there exist an integer
n
n
and a vector
u
∈
Z
l
u\in \mathbb {Z}^l
such that
u
+
p
i
(
n
)
v
i
∈
S
u+p_i(n)v_i\in S
for each
i
≤
k
i\le k
.