The Hilbert Function and the Hilbert Polynomial

Q: Is the Hilbert function itself eventually polynomial, or only its partial sums?

Both. If h_M agrees with χ_M for s ≥ N then _K M_s = Δ χ_M(s-1) for s > N , so the graded-piece Hilbert function is eventually the polynomial Δ χ_M(t-1) , one degree lower. The two statements are equivalent; the cumulative one is chosen here because it is the one that measures a filtration.

Q: Why is the Hilbert polynomial not required to have integer coefficients?

Because it cannot. Already t+22 = 12 t^2 + 32 t + 1 has non-integer coefficients while taking integer values at every integer. The correct integrality statement is that the coordinates in the basis t0, t1, … are integers.

Q: Does the theorem need the ground field to be algebraically closed, or infinite?

No. The proof is a dimension count over K and an induction on the number of variables; it works over any field, of any characteristic, finite or not. The dimension theory built on it inherits that generality; it is the later D-module theorems that need characteristic zero.

Q: Why does the degree drop by exactly one when I quotient by one homogeneous element?

If f is a non-zero-divisor of degree k , then χ_S/fS(t) = t+mm - t-k+mm , and subtracting a shift of a polynomial kills its leading term and leaves k ⋅ t^m-1/(m-1)! . Geometrically, cutting affine space by one non-trivial equation drops the dimension of the support by one. If f is a zero-divisor the drop can fail to happen at all.

Q: How does this connect to the dimension of a variety?

For a homogeneous ideal I , the degree of the cumulative Hilbert polynomial of S/I is the Krull dimension of S/I , that is, the dimension of the affine cone cut out by I . This is the statement that later identifies d(M) with the dimension of the characteristic variety.

Q: Can two non-isomorphic modules have the same Hilbert polynomial?

Easily. S/(y_1) and S/(y_2) are non-equal submodule quotients with identical Hilbert polynomials, and even non-isomorphic modules with the same annihilator can share one. The Hilbert polynomial is a numerical shadow; it sees size, not structure. Two of its coefficients survive into D-module theory, and even those do not determine a module.

Overview

Dimension for modules over the Weyl algebra is not defined directly. It is imported from commutative algebra along a route set up in the preceding chapters: a finitely generated $A_n$ -module is equipped with a good filtration with respect to the Bernstein filtration, its associated graded module is a finitely generated graded module over a commutative polynomial ring in $2n$ variables, and the growth of that graded module is measured. This page is about the measuring instrument.

The instrument is the Hilbert function. For a graded module $M = ⨁ _j ≥ 0 M_j$ over a polynomial ring, each component $M_j$ is a finite dimensional vector space, and the Hilbert function records those dimensions. What makes the invariant usable is a theorem of Hilbert: the accumulated dimensions $∑ _i=0^s _K M_i$ are not merely bounded by a polynomial in $s$ , they are equal to a polynomial in $s$ once $s$ is large enough. A counting problem with no visible algebraic structure turns out to have an exactly polynomial answer.

The reason to be slightly surprised is that nothing about the definition of a graded module suggests polynomiality. The module is presented by generators and relations of arbitrary degrees; the dimensions of its components are the ranks of a sequence of unrelated-looking linear systems. The theorem says those ranks are governed, from some point onwards, by finitely many rational numbers - the coefficients of $χ_M$ . Two of those numbers, the degree and the leading coefficient, are what become the dimension and the multiplicity of a Weyl module.

This page states the theorem, fixes the convention (the cumulative one - a real source of confusion with the commutative algebra literature), develops the arithmetic of numerical polynomials that the statement needs, and computes examples. The proof is given separately, on the existence proof page.

Definition

Throughout, $K$ is a field, $S = K[y_1, … ,y_m]$ is a polynomial ring with the standard grading in which every variable has degree $1$ , and $S_i$ denotes the span of the monomials of total degree $i$ . A graded P5-module is an $S$ -module $M$ with a decomposition $M = ⨁ _j M_j$ of $K$ -vector spaces such that $S_i M_j ⊂ eq M_i+j$ .

Hilbert function and cumulative Hilbert function

Let $M = ⨁ _j ≥ 0 M_j$ be a graded $S$ -module with all components finite dimensional over $K$ . The Hilbert function of $M$ is

H_M(s) = _K M_s, s ≥ 0,

and its cumulative Hilbert function is the partial sum

h_M(s) = ∑ _i=0^s _K M_i.

Numerical polynomial

A polynomial $p(t) ∈ Q[t]$ is numerical if $p(s) ∈ Z$ for every sufficiently large integer $s$ . Numerical polynomials need not have integer coefficients: $12t^2 + 12t$ is numerical, being $t+12$ .

Existence of the Hilbert polynomialCoutinho (9.1.1); Hilbert 1890

Let $M = ⨁ _j ≥ 0 M_j$ be a finitely generated graded module over $S = K[y_1, … ,y_m]$ . There exist a polynomial $χ_M(t) ∈ Q[t]$ and an integer $N ≥ 0$ such that

∑ _i=0^s _K M_i = χ_M(s) for every s ≥ N.

(9.1)

The polynomial $χ_M$ is uniquely determined by this property, is numerical, and has $χ_M ≤ m$ .

Why the components are automatically finite dimensional

A finitely generated graded module is generated by finitely many homogeneous elements $u_1, … ,u_r$ of degrees $e_1, … ,e_r$ . Then $M_j = ∑ _k S_j-e_k u_k$ is spanned by finitely many elements, so it is finite dimensional; and $M_j = 0$ for $j < _k e_k$ . The hypothesis $M_j = 0$ for $j < 0$ in the theorem is a normalisation, arranged by shifting degrees, and it is what makes the sum in (9.1) finite.

Core Concepts

Why the cumulative version is the right one here

In commutative algebra textbooks the Hilbert polynomial is usually attached to $H_M(s) = _K M_s$ directly. In D-module theory the cumulative version is used instead, and not for taste. If $Γ = Γ_i$ is a filtration of a vector space with $Γ_-1 = 0$ and $M = gr^Γ M = ⨁ _i Γ_i / Γ_i-1$ is its associated graded object, then the telescoping sum

∑ _i=0^s _K ( Γ_i / Γ_i-1 ) = _K Γ_s

says that the cumulative Hilbert function of the associated graded module is the honest dimension of the $s$ -th filtration piece. That is the quantity one actually wants to control: how fast the filtration of a module grows. The graded-piece Hilbert function would measure the growth of the successive quotients instead, which is one derivative away from the question.

Why a polynomial appears at all

The mechanism is visible in the free case. For $S$ itself, $_K S_i$ is the number of monomials of degree $i$ in $m$ variables, namely $i+m-1m-1$ , and summing gives $s+mm$ . Both are polynomials in the counting variable, because counting lattice points in a simplex is a polynomial problem. A finitely generated module is a quotient of a finite direct sum of shifted copies of $S$ , so its component dimensions are differences of such counts - corrected by the relations, and then by the relations among relations. The theorem says the correction terms are themselves eventually polynomial, and the induction that proves it is on the number of variables.

Numerical polynomials and the binomial basis

Because Hilbert functions are integer-valued while their polynomials have rational coefficients, the correct coefficient system is not $1, t, t^2, …$ but the binomial polynomials

tr = (t(t-1) ⋅ s(t-r+1))/(r!), t0 = 1.

These form a $ℚ$ -basis of $Q[t]$ in which a polynomial is numerical exactly when all its coordinates are integers, and in which the difference operator $Δ f(t) = f(t+1) - f(t)$ acts by shifting the index down: $Δ tr = tr-1$ . Working in this basis converts questions about integrality and about growth into bookkeeping.

What the degree measures

The degree of $χ_M$ is a coarse but robust measure of how big $M$ is. It is $m$ when $M$ has a free direct summand, $0$ when $M$ is finite dimensional over $K$ , and in general it equals the Krull dimension of the support of $M$ - the dimension of the closed subvariety of affine $m$ -space cut out by the annihilator of $M$ . That identification is standard commutative algebra rather than part of the Primer's development, but it is the reason the invariant deserves the name dimension once it is transported to the Weyl algebra.

Key Equations

The free module supplies the reference values. For $S = K[y_1, … ,y_m]$ with the standard grading,

_K S_i = i+m-1m-1, χ_S(t) = ∑ _i=0^t i+m-1m-1 = t+mm.

(9.2)

The second identity is the hockey-stick summation for binomial coefficients; it holds for every integer $t ≥ 0$ , with no exceptional range.

Expanding the leading behaviour of the right-hand side,

t+mm = (t^m)/(m!) + ((m+1))/(2,(m-1)!),t^m-1 + ⋅ s,

(9.3)

so $χ_S$ has degree $m$ and leading coefficient $1/m!$ . Shifting the grading shifts the argument: writing $S(-k)$ for the free module with generator in degree $k$ ,

χ_S(-k)(t) = t-k+mm.

(9.4)

Every graded short exact sequence of finitely generated graded modules, with degree-preserving maps,

0 → M' → M → M'' → 0 ⟹ χ_M(t) = χ_M'(t) + χ_M''(t),

(9.5)

because dimension is additive on exact sequences of vector spaces in each degree, and the partial sums inherit the additivity. Finally, in the binomial basis a numerical polynomial of degree $d$ is written

χ_M(t) = ∑ _r=0^d c_r tr, c_0, … ,c_d ∈ Z, c_d > 0,

(9.6)

and the leading coefficient of $χ_M$ as an ordinary polynomial is $c_d/d!$ , so that $d!$ times it is the integer $c_d$ .

Variable Definitions

$K$: the ground field; no characteristic assumption is needed for this page
$S$: the polynomial ring $K[y_1, … ,y_m]$ with the standard grading
$m$: the number of variables; it becomes $2n$ when $S$ is the associated graded ring of $A_n$ for the Bernstein filtration
$M_j$: the homogeneous component of degree $j$ of the graded module $M$
$H_M(s)$: the Hilbert function $_K M_s$
$h_M(s)$: the cumulative Hilbert function $∑ _i ≤ s _K M_i$
$χ_M(t)$: the Hilbert polynomial: the polynomial agreeing with $h_M(s)$ for all large $s$
$N$: the threshold beyond which $h_M$ and $χ_M$ agree
$Δ$: the difference operator, $Δ f(t) = f(t+1) - f(t)$
$(\binom{t}{r})$: the binomial polynomial of degree $r$ in the variable $t$

Properties and Behaviour

Uniqueness

The Hilbert polynomial is unique. Two polynomials agreeing at infinitely many integers are equal, so the condition in (9.1) pins $χ_M$ down completely, even though the threshold $N$ is not canonical.

Integrality and positivityCoutinho (9.1.2)(2)

$χ_M$ is a numerical polynomial, so it has the form (9.6) with integer coordinates in the binomial basis. If $M ≠ 0$ then $h_M$ is non-decreasing and eventually positive, so the leading coefficient of $χ_M$ is positive; consequently $d! ⋅ (leading coefficient)$ is a positive integer whenever $χ_M = d$ . This is exactly the statement that makes multiplicity an integer.

Degree bound

If $M$ is generated by $r$ homogeneous elements then $M$ is a quotient of $⨁ _k=1^r S(-e_k)$ , so by (9.4) and (9.5), $h_M(s) ≤ ∑ _k s-e_k+mm$ for all $s$ . Hence $χ_M ≤ m$ . Equality holds exactly when the annihilator of $M$ is the zero ideal, since by the remark below the degree is the dimension of the support.

Additivity

For a graded short exact sequence, $χ_M = χ_M' + χ_M''$ , so $χ_M = ( χ_M', χ_M'')$ : no cancellation can occur in the top degree because both leading coefficients are non-negative. This is the seed of the whole exact-sequence theory of dimension and multiplicity.

Degree and support

For a finitely generated graded $S$ -module $M$ , $χ_M$ equals the Krull dimension of $S/Ann(M)$ . The Primer does not prove this and does not need it, but it is the statement that later reappears geometrically: the dimension of a Weyl module equals the dimension of its characteristic variety.

Examples and Special Cases

The polynomial ring itself

For $M = S = K[y_1, … ,y_m]$ , $χ_S(t) = t+mm$ , of degree $m$ and leading coefficient $1/m!$ . Here $N = 0$ : the polynomial is correct from the very first value. With $m = 2n$ this computes $_K B_t = t+2n2n$ for the Bernstein filtration of $A_n$ , and hence $d(A_n) = 2n$ .

A hypersurface quotient

Let $f ∈ S$ be homogeneous of degree $k$ and $M = S/fS$ . Since $S$ is a domain, multiplication by $f$ is injective, giving the graded exact sequence $0 → S(-k) → S → M → 0$ and therefore

χ_M(t) = t+mm - t-k+mm,

a polynomial of degree $m-1$ with leading coefficient $k/(m-1)!$ . The dimension drops by one and the leading coefficient remembers the degree of $f$ . This single computation later delivers $d(A_n/A_n D) = 2n-1$ and multiplicity equal to the Bernstein degree of $D$ .

A finite dimensional module

If $_K M < ∈ fty$ then $M_j = 0$ for large $j$ , so $h_M(s) = _K M$ for $s ≫ 0$ and $χ_M$ is the constant $_K M$ , of degree $0$ . The Hilbert polynomial of a finite dimensional module is its dimension, and nothing else survives.

Where the threshold is genuinely needed

Take $m = 1$ , $S = K[y]$ , and $M = S ⊕ (S/y^5 S)$ . Then $h_M(s) = (s+1) + (s+1,5)$ , which equals $s+6$ only for $s ≥ 4$ . So $χ_M(t) = t+6$ and no smaller threshold than $N = 4$ works. Torsion pieces are precisely what postpone the agreement.

A module with two variables' worth of growth

In $S = K[y_1,y_2,y_3]$ take $M = S/(y_3)$ . Then $M ≅ K[y_1,y_2]$ as a graded ring, so $χ_M(t) = t+22$ , of degree $2 = 3 - 1$ , consistent with the hypersurface formula at $k = 1$ .

Worked Example

The Hilbert polynomial of $K[x,ξ]/(xξ)$

Step 1 - the module and why it is worth computing
Take $m = 2$ , $S = K[x,ξ]$ with $x = ξ = 1$ , and $M = S/(xξ)$ . This is not an arbitrary choice: $S$ is the associated graded ring of $A_1$ for the Bernstein filtration, and $x ξ$ is the symbol of the operator $x∂ - λ$ . The Hilbert polynomial computed here is therefore the one that will give the dimension and multiplicity of the $A_1$ -module $A_1/A_1(x∂ - λ)$ .
Step 2 - count each homogeneous component
A basis of $M_i$ is given by the monomials $x^a ξ^b$ with $a+b = i$ that are not divisible by $x ξ$ , that is, those with $a = 0$ or $b = 0$ . For $i = 0$ there is one such monomial, namely $1$ . For $i ≥ 1$ there are exactly two, namely $x^i$ and $ξ^i$ . Hence

$H_M(0) = 1, H_M(i) = 2 (i ≥ 1).$
Step 3 - accumulate
Summing, $h_M(s) = 1 + 2s$ for every $s ≥ 0$ . Check the first three values by listing monomials outright. For $s = 0$ : $1$ , so $h_M(0) = 1$ . For $s = 1$ : $1, x, ξ$ , so $h_M(1) = 3$ . For $s = 2$ : $1, x, ξ, x^2, ξ^2$ - the monomial $x ξ$ is excluded - so $h_M(2) = 5$ . The formula gives $1, 3, 5$ .
Step 4 - confirm by the exact sequence
Independently, $x ξ$ is a non-zero-divisor in the domain $S$ , so $0 → S(-2) → S → M → 0$ is exact and the hypersurface formula applies with $m = 2$ , $k = 2$ :

$χ_M(t) = t+22 - t2 = ((t+2)(t+1))/(2) - (t(t-1))/(2) = (4t+2)/(2) = 2t+1.$

The two computations agree, and here $N = 0$ .
Step 5 - read off the invariants
$χ_M(t) = 2t+1$ has degree $1$ and leading coefficient $2$ . In the binomial basis, $2t+1 = 2 t1 + 1 t0$ , with integer coordinates as the theory requires. The degree is $1$ and $1! × 2 = 2$ , so the associated Weyl module has dimension $1$ and multiplicity $2$ .

Result

$χ_M(t) = 2t + 1$ , valid for every $s ≥ 0$ : degree $1$ , leading coefficient $2$ . Transported to the Weyl algebra this says $d (A_1/A_1(x∂ - λ) ) = 1$ and $e (A_1/A_1(x∂ - λ) ) = 2$ - a holonomic module of multiplicity $2$ , not $1$ , so multiplicity genuinely carries information beyond dimension.

Applications and Industry Use

In a mathematics topic, this section covers downstream use inside mathematics, computing and engineering rather than a manufactured product.

Dimension and multiplicity of Weyl modules. The degree and normalised leading coefficient of $χ$ become $d(M)$ and $e(M)$ ; every result in Chapter 9 is a statement about Hilbert polynomials in disguise.
Degree and genus in algebraic geometry. For the homogeneous coordinate ring of a projective variety $X ⊂ eq P^m-1$ the Hilbert polynomial encodes $dim X$ , its degree, and its arithmetic genus; it is the discrete invariant that Hilbert schemes are indexed by.
Complexity of Gröbner bases. The Hilbert function of the ideal of leading terms equals that of the ideal itself, which turns Gröbner basis computations into Hilbert function computations and underlies Hilbert-driven Buchberger strategies.
Growth of algebras. Gelfand-Kirillov dimension, the noncommutative analogue used on the dimension page, is defined by exactly this growth-rate comparison; for $A_n$ the two notions coincide.
Coding and combinatorics. Counting monomials modulo an ideal is the same computation that counts standard monomials, evaluation codes and lattice points, so Hilbert functions appear far from D-modules.

Standards and Codes

For mathematics, the relevant standards are notation, numeric and markup standards together with reference implementations.

ISO 80000-2 fixes the notation used here: $ℚ$ and $ℤ$ for the rationals and integers, $(\binom{t}{r})$ for the binomial coefficient, and upright type for the operator names $dim$ and $deg$ .
ISO/IEC 40314 (MathML 3.0) is the encoding in which every expression on this page is delivered, so the mathematics is machine-readable rather than an image.
There is no standard for which convention the phrase "Hilbert polynomial" denotes. This collection always means the cumulative one, following Coutinho (9.1.1). When exchanging data with a computer algebra system, check: Macaulay2's hilbertPolynomial returns the non-cumulative projective Hilbert polynomial by default.
Coutinho cites results as chapter-and-item pairs; the existence theorem for the Hilbert polynomial is $(9.1.1)$ and the numerical polynomial lemma is $(9.1.2)$ .

Computational Notes

Read this as the manufacturing section of the template: how the object is actually built by machine, at what cost, and where the computation stops being decidable.

Computing $χ_M$ for an explicitly presented graded module is a solved problem in commutative computer algebra, and the standard route runs through leading terms.

Present $M$ as $⨁ _k S(-e_k) / N$ with $N$ generated by explicit homogeneous vectors.
Compute a Gröbner basis of $N$ for any term order; the module of leading terms $in(N)$ has the same Hilbert function as $N$ , because a Gröbner basis yields a monomial basis of the quotient.
Compute the Hilbert series of the monomial quotient by inclusion-exclusion over the generators of $in(N)$ , obtaining a rational function $Q(z)/(1-z)^m$ with $Q ∈ Z[z]$ .
Expand: the cumulative Hilbert polynomial is read off from $Q(z)/(1-z)^m+1$ , whose pole order at $z=1$ is $χ_M + 1$ and whose leading Laurent coefficient there determines the leading coefficient of $χ_M$ after division by $( χ_M)!$ .

In practice one calls a library rather than doing this by hand: Macaulay2 provides hilbertSeries and hilbertPolynomial, Singular has hilb, and SageMath exposes both through its interfaces. The cost is dominated by the Gröbner basis step, which is doubly exponential in the number of variables in the worst case, though Hilbert-driven strategies use a known Hilbert series to prune the computation dramatically.

A cheap correctness check

Whatever the implementation, the first few values of $h_M$ can be computed by brute-force linear algebra in each degree. Any disagreement between brute force and the fitted polynomial above the reported threshold is a bug; disagreement below it is expected and means nothing.

Limits of Validity

The theorem is about a graded module over a commutative polynomial ring with the standard grading, and each of those words is doing work.

Standard grading. If the variables are given different positive weights the counting function is still eventually quasi-polynomial, but it need not be polynomial: it can be polynomial on residue classes modulo the least common multiple of the weights. This is exactly why the order filtration, in which $x_i$ has degree $0$ , cannot be fed into this theorem as it stands - its graded pieces are not even finite dimensional over $K$ .
Commutativity. The theorem is applied to $gr^B A_n ≅ K[x_1, … ,x_n,ξ_1, … ,ξ_n]$ , which is commutative even though $A_n$ is not. Commutativity of the associated graded ring is the whole reason the transfer works.
Finite generation. Without it, growth can be arbitrary. Finite generation of $gr^Γ M$ is precisely the definition of a good filtration, which is why good filtrations are the ones the dimension theory uses.
Eventual, not universal. The conclusion is an equality for $s ≥ N$ ; nothing is claimed below $N$ , and $N$ is not bounded in terms of $m$ alone.

Remark

Nothing here needs characteristic zero. Characteristic zero enters the D-module story later, through the simplicity of P0 and hence through Bernstein's inequality, not through the Hilbert polynomial.

Failure Modes and Common Mistakes

Mixing the two conventions and losing a degree

The commutative algebra literature attaches the Hilbert polynomial to $_K M_s$ ; this collection, following the Primer, attaches it to the partial sums. The degrees differ by exactly one. Quoting "the Hilbert polynomial of $K[y_1, … ,y_m]$ has degree $m-1$ " inside the D-module dimension theory produces $d(A_n) = 2n-1$ and wrecks Bernstein's inequality. Always check which convention a formula is written in before importing it.

Assuming the polynomial is correct from $s = 0$

It often is - for $S$ and for hypersurface quotients the agreement is exact from the start - but not always. The module $S ⊕ S/y^5S$ over $K[y]$ needs $s ≥ 4$ . Any argument that evaluates a Hilbert polynomial at a small argument, or that compares two of them at one point, is invalid. Comparisons must be asymptotic.

Expecting integer coefficients

$χ_S(t) = t+mm$ has leading coefficient $1/m!$ . The polynomial is integer-valued, not integer-coefficiented. This is why multiplicity is defined as $d!$ times the leading coefficient rather than as the leading coefficient itself: the factorial is precisely what clears the denominator.

Applying the theorem to a module that is not finitely generated or not bounded below

Both hypotheses are load bearing. The graded module $⨁ _j ≥ 0 K$ over $K[y]$ with $y$ acting as zero is not finitely generated and has $h(s) = s+1$ - accidentally polynomial - but $⨁ _j ≥ 0 K^j!$ is a graded $K[y]$ -module with $y$ acting as zero whose growth is faster than any polynomial. If the grading is unbounded below, as for $K[y,y^-1]$ , the partial sum in (9.1) is not even defined.

Historical Notes

Hilbert introduced the function now named after him in his 1890 paper on the theory of algebraic forms, in the course of proving the syzygy theorem. The point of the finite free resolution was precisely to make the counting function computable: an alternating sum over a resolution by shifted free modules turns the count into a manageable combination of binomial coefficients, which is where the eventual polynomiality comes from.

The interpretation of the degree as a dimension, and of the normalised leading coefficient as a multiplicity, matured in the first half of the twentieth century in the hands of van der Waerden, Krull and Samuel, and became the standard local dimension theory of Noetherian rings. By the time D-module theory needed it, it was textbook material.

Its use on the Weyl algebra is due to Bernstein in 1971-72. The observation that makes it work - that the associated graded ring of $A_n$ for the Bernstein filtration is an honest commutative polynomial ring in $2n$ variables with the standard grading, so that Hilbert's theorem applies verbatim - is a small one, and it is the hinge of the entire subject.

Comparison

Three closely related objects are all called "the Hilbert something" and they are routinely confused. The table fixes them for $M = S = K[y_1, … ,y_m]$ .

Three counting devices for a graded module, compared on the free module of rank one.
Object	Definition	Value for $S$	Degree in $t$
Hilbert function	$_K M_s$	$s+m-1m-1$	$m-1$
Cumulative Hilbert function (used here)	$∑ _i ≤ s _K M_i$	$s+mm$	$m$
Hilbert series	$∑ _s ( _K M_s),z^s$	$(1-z)^-m$	not a polynomial
Hilbert polynomial (this page)	polynomial equal to the cumulative function for $s ≫ 0$	$t+mm$	$m$

Key Takeaways

Key points

For a finitely generated graded module $M$ over $K[y_1, … ,y_m]$ with $M_j = 0$ for $j<0$ , the accumulated dimensions $∑ _i ≤ s _K M_i$ agree with a polynomial $χ_M(s)$ for all large $s$ .
The cumulative convention is used because $∑ _i ≤ s _K(Γ_i/Γ_i-1) = _K Γ_s$ , the growth of a filtration.
$χ_M$ is numerical: rational coefficients, integer values, integer coordinates in the binomial basis $(\binom{t}{r})$ .
$χ_S(t) = t+mm$ for the polynomial ring itself, of degree $m$ and leading coefficient $1/m!$ .
$χ_M ≤ m$ , and $χ_M$ equals the dimension of the support of $M$ .
Degree and $d!$ times the leading coefficient are what become the dimension and multiplicity of a module over the Weyl algebra.
The equality is eventual: nothing is claimed below the threshold $N$ .

FAQs

Is the Hilbert function itself eventually polynomial, or only its partial sums?

Both. If $h_M$ agrees with $χ_M$ for $s ≥ N$ then $_K M_s = Δ χ_M(s-1)$ for $s > N$ , so the graded-piece Hilbert function is eventually the polynomial $Δ χ_M(t-1)$ , one degree lower. The two statements are equivalent; the cumulative one is chosen here because it is the one that measures a filtration.

Why is the Hilbert polynomial not required to have integer coefficients?

Because it cannot. Already $t+22 = 12 t^2 + 32 t + 1$ has non-integer coefficients while taking integer values at every integer. The correct integrality statement is that the coordinates in the basis $t0, t1, …$ are integers.

Does the theorem need the ground field to be algebraically closed, or infinite?

No. The proof is a dimension count over $K$ and an induction on the number of variables; it works over any field, of any characteristic, finite or not. The dimension theory built on it inherits that generality; it is the later D-module theorems that need characteristic zero.

What is the threshold

N

in practice?

For the modules that arise from good filtrations on cyclic Weyl modules it is usually $0$ or very small. In general $N$ is related to the regularity of $M$ , which can be bounded in terms of the degrees of a generating set and of the relations, but the bounds are large. No result in this collection depends on the size of $N$ , only on its existence.

Why does the degree drop by exactly one when I quotient by one homogeneous element?

If $f$ is a non-zero-divisor of degree $k$ , then $χ_S/fS(t) = t+mm - t-k+mm$ , and subtracting a shift of a polynomial kills its leading term and leaves $k ⋅ t^m-1/(m-1)!$ . Geometrically, cutting affine space by one non-trivial equation drops the dimension of the support by one. If $f$ is a zero-divisor the drop can fail to happen at all.

How does this connect to the dimension of a variety?

For a homogeneous ideal $I$ , the degree of the cumulative Hilbert polynomial of $S/I$ is the Krull dimension of $S/I$ , that is, the dimension of the affine cone cut out by $I$ . This is the statement that later identifies $d(M)$ with the dimension of the characteristic variety.

Can two non-isomorphic modules have the same Hilbert polynomial?

Easily. $S/(y_1)$ and $S/(y_2)$ are non-equal submodule quotients with identical Hilbert polynomials, and even non-isomorphic modules with the same annihilator can share one. The Hilbert polynomial is a numerical shadow; it sees size, not structure. Two of its coefficients survive into D-module theory, and even those do not determine a module.

Does the Primer prove the identification of the degree with the dimension of the support?

No, and it does not need to. Coutinho's Chapter 9 defines dimension as the degree of the Hilbert polynomial and works with that definition. The geometric identification appears later, in the characteristic variety chapter, and is treated there as a separate theorem.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 9 §1, theorem (9.1.1) and lemma (9.1.2).
D. Hilbert, Über die Theorie der algebraischen Formen, Mathematische Annalen 36 (1890), 473-534 - the origin of the Hilbert function and the syzygy theorem.
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969 - Ch. 11, for the Hilbert-Samuel theory in the non-cumulative convention.
D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer, 1995 - Ch. 12, for Hilbert functions, degree and dimension.
J.-E. Björk, Rings of Differential Operators, North-Holland Mathematical Library 21, North-Holland, 1979 - Ch. 1, for the filtered-to-graded transfer used here.
G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, revised edition, Graduate Studies in Mathematics 22, American Mathematical Society, 2000.
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry - documentation for hilbertSeries and hilbertPolynomial.
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization.
ISO/IEC 40314:2016, Information technology - Mathematical Markup Language (MathML) Version 3.0, International Organization for Standardization.

AI Suggested Questions

Compute the cumulative Hilbert polynomial of $K[y_1,y_2,y_3]/(y_1y_2, y_1y_3)$ and identify its degree and leading coefficient.
Show that $Δ tr = tr-1$ directly from Pascal's rule, and deduce that the binomial polynomials are a basis adapted to the difference operator.
Give a finitely generated graded module whose Hilbert polynomial has threshold $N = 10$ .
Explain why the Hilbert function of an ideal equals that of its ideal of leading terms with respect to any term order.
Work out the Hilbert polynomial of $K[x,ξ]/(x^2ξ)$ and compare it with the case $(xξ)$ .
Verify that the two conventions for the Hilbert polynomial differ by exactly one in degree, and say which one Macaulay2 returns.
Show that a graded module over a polynomial ring with weighted variables can have a quasi-polynomial rather than polynomial counting function.

Related Calculators

Hilbert Polynomial Calculator
Fit a numerical polynomial to the first values of a cumulative Hilbert function and report degree and leading coefficient.
Calculator
Binomial Basis Expansion
Rewrite a numerical polynomial in the basis of binomial polynomials and check that its coordinates are integers.
Calculator