In this paper we consider sums of squares of vector fields in
R
2
\mathbb {R}^2
satisfying Hörmander’s condition and with polynomial, but non-(quasi-)homoge- neous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is
\[
P
=
X
2
+
Y
2
+
∑
j
=
1
L
Z
j
2
,
P=X^2+Y^2+\sum _{j=1}^{L}Z_j^2,
\]
with
\[
X
=
D
x
,
Y
=
a
0
(
x
,
y
)
x
q
−
1
D
y
,
Z
j
=
a
j
(
x
,
y
)
x
p
j
−
1
y
k
j
D
y
,
X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y,
\]
with
a
j
(
0
,
0
)
≠
0
a_{j}(0, 0) \neq 0
,
j
=
0
,
1
,
…
,
L
j = 0, 1, \ldots , L
and
q
>
p
j
,
{
k
j
}
q>p_j, \{k_j\}
arbitrary. The theorem we prove is that
P
P
is Gevrey-s hypoelliptic for
s
≥
1
1
−
T
,
T
=
max
j
q
−
p
j
q
k
j
.
s\geq \frac {1}{1-T}, T = \max _j \frac {q-p_j}{q k_j}.