We prove, among other results, that the one-dimensional local domain
A
A
is Henselian if and only if for every maximal ideal
M
M
in the Laurent polynomial ring
A
[
T
,
T
−
1
]
A[T,{T^{ - 1}}]
, either
M
∩
A
[
T
]
M \cap A[T]
or
M
∩
A
[
T
−
1
]
M \cap A[{T^{ - 1}}]
is a maximal ideal. The discrete valuation ring
A
A
is Henselian if and only if every pseudoWeierstrass polynomial in
A
[
T
]
A[T]
is Weierstrass. We apply our results to the complete intersection problem for maximal ideals in regular Laurent polynomial rings.