A lex-plus-powers ideal for integers 
in the polynomial
ring ![S=k[x_1,...,x_n]](/images/equations/Lex-Plus-PowersIdeal/Inline2.svg)
,
over a field 
, with 
, is a monomial ideal 
containing 
such that, in each total
degree, its monomials outside 
form an initial segment of the decreasing lexicographic
order with 
.
Thus, whenever 
,
every lexicographically larger monomial of the same
total degree outside 
also belongs to 
. The Hilbert function
of 
fixes the size of each segment and therefore determines 
for the prescribed powers.
For example, 
is lex-plus-powers for 
and 
. In total degree 2, the
available monomials are 
, and 
contains the first of these. Its quotient Hilbert
function has values 1, 2, 1, 0, ....
