The power series ring ![k[[x_1,...,x_d]]](/images/equations/PowerSeriesRing/Inline1.svg)
over a field 
is the ring
of formal power series in the variables 
. Addition
is coefficientwise, and multiplication uses the
usual coefficient-convolution rule. Each product coefficient
involves only finitely many summands, so no analytic convergence is required.
This is a complete local ring with maximal ideal 
and residue field 
. It is a regular local ring
of Krull dimension 
(Bruns and Herzog 1998). The quotient by 
retains exactly the monomials
of total degree below 
, which gives Hilbert-Samuel
multiplicity 1.
