In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and in particular in the field of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, a Hilbert–Poincaré series (also known under the name
Hilbert series
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homog ...
), named after
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
and
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
, is an adaptation of the notion of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
to the context of
graded algebraic structures (where the dimension of the entire structure is often infinite). It is a
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
in one indeterminate, say
, where the coefficient of
gives the dimension (or rank) of the sub-structure of elements homogeneous of degree
. It is closely related to the
Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homog ...
in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
of its argument
.
Definition
Let ''K'' be a field, and let
be an
-
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a ''grading'' or ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.
For ...
over ''K'', where each subspace
of vectors of degree ''i'' is finite-dimensional. Then the Hilbert–Poincaré series of ''V'' is the
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
:
A similar definition can be given for an
-graded ''R''-module over any
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''R'' in which each submodule of elements homogeneous of a fixed degree ''n'' is
free of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the Hilbert–Poincaré series is considered has additional structure, for instance, that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure.
Example: Since there are
monomials of degree ''k'' in variables
(by induction, say), one can deduce that the sum of the Hilbert–Poincaré series of