In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set
E
:=
{
z
:
|
z
|
≤
r
E:=\big \{z\,:\, |z|\le r
and
|
P
(
z
)
|
≤
ϵ
n
}
|P(z)|\le \epsilon ^{n}\big \}
, for a polynomial
P
P
of degree
≤
n
\le n
. Usually,
P
P
is taken to be monic, and either Cartan’s Lemma or potential theory is used to estimate the size of
E
E
, in terms of Hausdorff contents, planar Lebesgue measure
m
2
m_{2}
, or logarithmic capacity cap. Here we normalize
‖
P
‖
L
∞
(
|
z
|
≤
r
)
=
1
\|P\|_{L_{\infty }\bigl (|z|\le r\bigr )}=1
and show that cap
(
E
)
≤
2
r
ϵ
(E)\le 2r\epsilon
and
m
2
(
E
)
≤
π
(
2
r
ϵ
)
2
m_{2} (E)\le \pi (2r\epsilon )^{2}
are the sharp estimates for the size of
E
E
. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on
C
n
\mathbb {C}^{n}
or product capacity and Favarov’s capacity. Several of our estimates are sharp with respect to order in
r
r
and
ϵ
\epsilon
.