We prove that all polynomials in several variables can be decomposed as the sums of
k
k
th powers:
P
(
x
1
,
…
,
x
n
)
=
Q
1
(
x
1
,
…
,
x
n
)
k
+
⋯
+
Q
s
(
x
1
,
…
,
x
n
)
k
P(x_1,\ldots ,x_n) = Q_1(x_1,\ldots ,x_n)^k+\cdots + Q_s(x_1,\ldots ,x_n)^k
, provided that elements of the base field are themselves sums of
k
k
th powers. We also give bounds for the number of terms
s
s
and the degree of the
Q
i
k
Q_i^k
. We then improve these bounds in the case of two-variable polynomials of large degree to get a decomposition
P
(
x
,
y
)
=
Q
1
(
x
,
y
)
k
+
⋯
+
Q
s
(
x
,
y
)
k
P(x,y) = Q_1(x,y)^k+\cdots + Q_s(x,y)^k
with
deg
Q
i
k
⩽
deg
P
+
k
3
\deg Q_i^k \leqslant \deg P + k^3
and
s
s
that depends on
k
k
and
ln
(
deg
P
)
\ln (\deg P)
.