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, ....
