A Roadmap Through Algebraic D-module Theory

Q: What is the shortest route to understanding holonomic modules?

Canonical form and simplicity of A_n ; the polynomial module as the standard example; filtered and graded modules; good filtrations; the Hilbert polynomial; Bernstein's inequality; then the definition. That is roughly seven chapters' worth of material and none of it requires geometry. The operations and the characteristic variety can both be deferred.

Overview

The elementary theory of algebraic D-modules has a clean shape, and knowing it in advance saves a great deal of time. It divides into two halves of roughly equal length. The first half attaches invariants to a module over the Weyl algebra $A_n$ : a dimension, a multiplicity, a characteristic variety. The second half constructs operations that turn modules over one Weyl algebra into modules over another, following a polynomial map between the underlying affine spaces. The two halves meet in a single theorem: the operations preserve the class of modules picked out by the invariants.

Before either half can begin, the ring itself has to be understood. The opening chapters establish that $A_n$ is a simple Noetherian domain with a canonical form for its elements, and place it inside the wider family of rings of differential operators. That material is short, entirely elementary, and used everywhere afterwards; nothing later can be read without it.

The pay-off comes in two application chapters at the end, which are deliberately unlike each other. One uses a D-module argument to attack the global asymptotic stability of polynomial vector fields, a question in dynamics. The other turns the finiteness properties of holonomic modules into an algorithm that proves combinatorial identities automatically. Between them they demonstrate that the machinery is not self-referential.

The order of the material is almost, but not quite, linear. Three parts of the theory are optional in the strict sense that nothing else depends on them, and identifying those in advance is the single most useful thing a roadmap can do. This page marks them, sets out the milestone theorems, and follows one small module all the way through the theory so that the architecture is visible in a single example.

Core Concepts

Four ideas structure the whole development, and each is introduced exactly once.

The ring is small enough to control

$A_n$ has a canonical form: every operator is uniquely $∑ c_αβx^α∂^β$ . That single fact makes it a domain, makes it Noetherian, and makes it simple. Simplicity is used constantly, because it forces every non-zero module to be faithful. Everything in the first two chapters exists to establish these properties, and they are never proved again.

Filtrations convert noncommutative questions into commutative ones

$A_n$ is not commutative, but it becomes so in the limit: filter it by degree and the associated graded ring is a polynomial ring in $2n$ variables. A finitely generated module carries a good filtration, its associated graded module is a finitely generated module over that polynomial ring, and commutative Hilbert-function theory applies. This is the mechanism behind the entire invariant theory. Chapters 7 and 8 build the mechanism; Chapter 9 uses it.

One number decides everything

The dimension $d(M)$ , the degree of the Hilbert polynomial, is confined by Bernstein's inequality to $n ≤ d(M) ≤ 2n$ . Modules attaining the minimum are holonomic, and they turn out to have every property one could want: finite length, cyclicity, closure under extension. A large fraction of the theory consists of proving that some naturally arising module is holonomic, because that single word carries all the finiteness.

Geometry acts through functors

A polynomial map $f : K^n → K^m$ does not map $A_m$ to $A_n$ , so operations on modules must be built by hand, out of tensor products. The inverse image generalises composing a function with $f$ ; the direct image generalises integrating over the fibres. Both are constructed by factoring an arbitrary map into a closed embedding followed by a projection, which is why embeddings get a chapter of their own.

Key Equations

Six formulas recur throughout. Everything on this page can be located by which of them it is about.

[∂_i, x_j] = δ_ij, D = ∑ _α,β c_αβ, x^α ∂^β,

(R.1)

the defining relation and the canonical form (Chs. 1-2).

gr, A_n ; ≅ ; K[x_1, … ,x_n,ξ_1, … ,ξ_n],

(R.2)

the associated graded ring, a commutative polynomial ring (Ch. 7).

_K Γ_m = χ_Γ(m) (m ≫ 0), d(M) = χ_Γ, e(M) = d(M)!, ⋅ ,[leading coefficient],

(R.3)

the Hilbert polynomial and the two numerical invariants (Ch. 9).

n ≤ d(M) ≤ 2n, M holonomic ⟺ M = 0 or d(M) = n,

(R.4)

Bernstein's inequality and the definition it licenses (Chs. 9-10).

Ch(M) = V (J(M) ) ⊂ eq K^2n, Ch(M) = d(M),

(R.5)

the characteristic variety and the geometric reading of the dimension (Ch. 11).

f^*M = K[x] ⊗ _K[y] M, f_+N = (direct image, built from f^* by side changing),

(R.6)

the two operations, in outline (Chs. 14-16).

Variable Definitions

$K$: the ground field, of characteristic zero throughout
$A_n$: the $n$ -th Weyl algebra over $K$
$D(R)$: the ring of differential operators of a commutative $K$ -algebra $R$
$B_m$ , $F_m$: the $m$ -th pieces of the Bernstein and order filtrations of $A_n$
$Γ_m$: the $m$ -th piece of a good filtration of a module
$gr$: the associated graded ring or module of a filtered object
$χ_Γ$: the Hilbert polynomial of a good filtration
$d(M)$ , $e(M)$: the dimension and the multiplicity of $M$
$Ch(M)$ , $J(M)$: the characteristic variety and the characteristic ideal of $M$
$f^*$ , $f_+$: the inverse and direct image functors attached to a polynomial map $f$

Properties and Behaviour

Five theorems carry the structural weight. Everything else is either preparation for one of them or a consequence of one of them.

Simplicity of the Weyl algebraCoutinho, Ch. 2 §2

For $K$ of characteristic zero, the only two-sided ideals of $A_n(K)$ are $0$ and $A_n$ . Consequence: every non-zero module is faithful, which is the hypothesis every dimension estimate silently uses.

Existence of the Hilbert polynomialCoutinho, Ch. 9 §1-§2

If $M$ is finitely generated with a good filtration $Γ$ , then $_K Γ_m$ agrees with a polynomial in $m$ for large $m$ , and its degree and leading coefficient are independent of the good filtration chosen. Consequence: $d(M)$ and $e(M)$ are invariants of $M$ , not of the presentation.

Bernstein's inequalityCoutinho (9.4.2)

$n ≤ d(M) ≤ 2n$ for every non-zero finitely generated $A_n$ -module. Consequence: the definition of holonomicity is not vacuous, and 'minimal dimension' is a meaningful condition.

Kashiwara's theoremCoutinho, Ch. 17 and Ch. 18 §3

For the embedding of a coordinate subspace $i : K^r ↪ K^n$ , the direct image $i_+$ is an equivalence between $A_r$ -modules and those $A_n$ -modules supported on the subspace. Consequence: modules concentrated on a subvariety are completely described by modules on that subvariety, which is what makes induction on dimension possible.

Preservation of holonomyCoutinho, Ch. 18

If $M$ is holonomic then so are P1 and P2, for any polynomial map $f$ . Consequence: the holonomic modules form a category closed under the geometric operations, which is the property that makes the applications possible at all.

Holonomy is preserved; dimension is not

It is worth stating explicitly, because it surprises everyone: the operations do not preserve $d(M)$ in general. A module of dimension strictly between $n$ and $2n$ can have an image of quite different dimension, and even the base ring changes, so the two dimensions are measured against different values of $n$ . What survives is the extremal condition $d(M) = n$ . The class is preserved even though the number is not.

Worked Example

One module, carried through the whole theory

Step 1 (Chs. 5-6) - present the module
Take $n = 1$ and $M = K[x]$ , with $x$ acting by multiplication and $∂$ by $d/dx$ . It is cyclic, generated by the constant $1$ , and the operators killing $1$ are exactly the multiples of $∂$ , so

$M ; ≅ ; A_1/A_1∂.$

Checking that this really defines an action, rather than merely writing down symbols, is the content of defining a module by generators and relations. The class of $x^a$ corresponds to $x^a ⋅ 1$ , and these are a $K$ -basis.
Step 2 (Chs. 7-9) - filter it and compute the invariants
Use the good filtration $Γ_m = B_m 1$ generated by the generator. Because $∂ 1 = 0$ , only the monomials $x^a$ with $a ≤ m$ survive, so $Γ_m$ is the space of polynomials of degree at most $m$ and

$_K Γ_m = m+1 for all m ≥ 0.$

Check by hand: $Γ_0 = K$ , dimension $1$ ; $Γ_1 = K + Kx$ , dimension $2$ ; $Γ_2$ adds $x^2$ , dimension $3$ . Hence $χ(m) = m+1$ , so $d(M) = 1$ and $e(M) = 1! ⋅ 1 = 1$ .
Step 3 (Chs. 10-11) - classify it
Since $n = 1$ and $d(M) = 1$ , $M$ is holonomic, and its length is at most $e(M) = 1$ , so $M$ is simple. Geometrically, with respect to the order filtration the associated graded module is $K[x]$ with $ξ$ acting as zero, so the characteristic ideal is generated by $ξ$ and

$Ch(M) = (x,ξ) ∈ K^2 : ξ = 0,$

the zero section of the cotangent space. It has dimension $1 = d(M)$ , as (R.5) requires, and it is Lagrangian - the geometric form of holonomicity.
Step 4 (Chs. 16-17) - push a module forward and compare
Now let $i : 0 ↪ K$ be the inclusion of the origin, and push forward the one-dimensional module over $A_0 = K$ . The direct image is

$i_+K ; ≅ ; A_1/A_1x,$

the delta module, with $K$ -basis the classes of $∂^b$ for $b ≥ 0$ . Filtering as before gives $_K Γ_m = m+1$ again, so $d = 1$ and $e = 1$ : holonomic and simple, exactly as (R.4) and the preservation theorem predict. Its characteristic variety is $x = 0$ , the cotangent fibre over the origin - a different Lagrangian, of the same dimension.
Step 5 (Ch. 18) - read off what was general
Nothing in Step 4 used the specific module: Kashiwara's theorem says that every $A_1$ -module supported at the origin arises as $i_+$ of a $K$ -vector space, and the preservation theorem says that holonomicity would have survived any polynomial map, not just this embedding. The example is the general theory shrunk to its smallest non-trivial instance.

Result

$K[x] ≅ A_1/A_1∂$ has $d = 1$ , $e = 1$ , is simple and holonomic, and has the zero section as characteristic variety. Its counterpart $A_1/A_1x$ , the direct image of a point, has the same invariants and the conormal fibre at the origin as characteristic variety. The two are not the only simple holonomic $A_1$ -modules - $A_1/A_1(∂ - 1)$ is another, with the same characteristic variety as $K[x]$ - but between them they illustrate every stage of the theory: presentation, filtration, invariants, geometry, operations, and preservation.

Applications and Industry Use

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

The two application chapters are chosen to be as unlike each other as possible, which is the point: they show that holonomicity is not an internal convenience of the theory.

Global stability of polynomial vector fields. Whether a polynomial vector field whose Jacobian has everywhere-negative eigenvalues must have a globally attracting equilibrium is a question in dynamics with no visible algebra in it. The key step admits a short D-module proof due to van den Essen, and the planar case is settled; see the global stability problem.
Automatic proof of identities. A definite sum or integral of a holonomic integrand satisfies a holonomic recurrence, which a computer can find and verify. This converts the proof of a binomial identity into a finite computation; see creative telescoping.
The Jacobian conjecture. The equivalence with the Dixmier conjecture shows that a famous open problem about polynomial maps is a problem about endomorphisms of $A_n$ - a translation in the opposite direction to the rest of the theory.
Analytic continuation. The b-function and the meromorphic continuation of $f^s$ were the historical motivation and remain the standard advertisement for the theory.
Beyond the elementary theory. Representation theory via localisation, perverse sheaves, and the Riemann-Hilbert correspondence all lie past the boundary of this collection but are reached by exactly the route mapped here.

Design Considerations

For a mathematical object, design considerations are the modelling choices: which ring, which filtration, which category to work in.

The order is nearly linear, and the exceptions are worth knowing before starting rather than discovering halfway through.

What can be skipped, and what cannot

Chapters 1, 2, 5, 7, 8, 9, 10 form the unavoidable spine. Nothing later makes sense without them.
Chapter 11 (characteristic varieties) depends on everything before it but is used nowhere afterwards in the elementary theory. It supplies the geometric interpretation of the dimension, and it is where algebraic geometry is required. Skipping it costs intuition, not logical continuity.
Chapter 4 (the Jacobian conjecture) is an application, but its first section on derivations of the polynomial ring reappears when inverse images are constructed, so it cannot be skipped entirely.
Chapter 6 (differential equations) is motivational for the invariant theory and essential for the applications; a reader interested only in ring theory can postpone it.
The categorical section of Chapter 18 restates the operations in the language of functors and is not used elsewhere; it is the natural bridge to the advanced literature.

Three reading paths

A reader who wants the algorithms should go 1, 2, 5, 7-10, then straight to the holonomic-function machinery, skipping the operations entirely: Zeilberger's method needs closure properties, not functors. A reader who wants the geometry should go 1-3, 5, 7-11, then 12-18, treating Chapter 11 as the goal rather than a detour. A reader who wants the ring theory should go 1-4 and 8, where the Dixmier conjecture and the automorphism group live, and can stop there.

Prerequisites, honestly assessed

Linear algebra and a first course in rings and modules suffice for Chapters 1 to 10. Chapter 11 needs affine algebraic geometry at the level of a first chapter of a standard text: varieties, ideals, dimension. The stability application needs the basic theory of ordinary differential equations, including linearisation at an equilibrium. Analysis is not required for any proof, but the examples are much flatter without it, since the interesting target modules are spaces of functions.

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.

A useful way to read the roadmap is to ask, at each stage, what is actually computable. The answer changes sharply between the two halves.

Chapters 1-2: arithmetic in $A_n$ is completely mechanical - rewrite to canonical form using the commutation relation. Every computer algebra system does this.
Chapters 7-9: computing $d(M)$ and $e(M)$ from a presentation requires a Gröbner basis in the Weyl algebra, then a commutative Hilbert polynomial. Feasible for small $n$ and modest degrees.
Chapter 10: deciding holonomicity is the same computation followed by a comparison with $n$ . Finding the b-function of a polynomial is much harder and is a specialised algorithm.
Chapter 11: computing a characteristic variety is a Gröbner basis calculation with respect to the order filtration, followed by a primary decomposition.
Chapters 14-18: inverse and direct images are computable in principle by restriction and integration algorithms, but the intermediate objects grow quickly, and this is where implementations meet their practical limits.

The packages that implement these steps are Dmodules in Macaulay2, dmod.lib and bfun.lib in Singular, ore_algebra in SageMath, and HolonomicFunctions.m in Mathematica. The identity-proving applications of Chapter 20 are the most robust in practice, precisely because they avoid the image functors.

Limits of Validity

The map above is a map of the elementary theory, and it stops well short of the modern subject. What lies outside it should be stated plainly.

No sheaves. Everything is affine, so a module over a single ring suffices. On a projective or general variety, D-modules are sheaves and the operations are defined only in the derived category.
No derived functors. The direct image is treated as a single functor; in the full theory it is a complex, and the higher terms carry essential information. Kashiwara's theorem in the form given here is the shadow of a statement about derived categories.
No regularity, no Riemann-Hilbert. The distinction between regular and irregular singularities is invisible at this level, and with it the correspondence between holonomic modules and constructible sheaves; see the outlook page.
Characteristic zero only. Every dimension statement fails in characteristic $p$ .
Smooth affine base. On singular varieties the ring of differential operators can fail even to be Noetherian, and none of the invariant theory can be assumed.

The boundary is where the elementary methods stop, not where the mathematics does

Each restriction above is a restriction on the methods, chosen so that the arguments stay at the level of linear algebra and commutative Hilbert theory. The corresponding general statements are mostly true, and mostly hard. A reader who finishes this route has the vocabulary for the general theory and none of its machinery, which is the right place to be before starting on it.

Failure Modes and Common Mistakes

Expecting the operations to preserve dimension

They do not, and the theory would be much easier if they did. Only the extremal case is preserved: a holonomic module has holonomic images. Since $f$ changes the number of variables, even the ambient bound $n ≤ d ≤ 2n$ is measured against a different $n$ before and after. Any argument that tracks $d(M)$ through an inverse or direct image must justify each step separately.

Starting at Chapter 11

The characteristic variety is the most appealing idea in the elementary theory, and it is the one that depends on the most. It needs good filtrations, the associated graded ring, the independence of the dimension from the filtration, and affine algebraic geometry. Reading it early produces a plausible-sounding but unfounded picture, and in particular the identity $Ch(M) = d(M)$ looks like a definition when it is a theorem.

Treating the two filtrations as interchangeable

The Bernstein filtration is used for the dimension theory because its pieces are finite dimensional over $K$ ; the order filtration is used for the geometry because its graded ring is the coordinate ring of the cotangent space. That the dimension comes out the same is a result, and it is proved after both are introduced. Substituting one for the other inside a proof is the most frequent technical error in the subject.

Reading the applications as consequences of a single theorem

Neither application follows formally from the preservation theorem. The stability argument uses a specific module built from the vector field and a specific finiteness statement about it; the identity-proving algorithm uses closure properties of holonomic functions, which are a corollary of the module theory but need their own translation. The theory supplies the tools, not the theorems.

Assuming every module has a dimension

Dimension is defined only for finitely generated modules, because only they carry good filtrations. Infinitely generated modules occur naturally - the field of rational functions in one variable is one - and for them $d(M)$ is simply not defined. Statements like 'every $A_n$ -module has dimension at least $n$ ' are false as written.

Historical Notes

The order of the material is not the order of discovery. The ring-theoretic results of Chapters 1 and 2 were found by Littlewood in 1933, long before anyone considered modules over the algebra for their own sake. The invariant theory of Chapters 7 to 10 is Bernstein's, from 1971-72, and was created for a specific purpose: to prove the meromorphic continuation of $f^s$ without resolution of singularities. The operations of Chapters 13 to 18 come from the analytic theory of Sato's school and Kashiwara's 1970 thesis, and were transported into the algebraic setting afterwards.

The geometric interpretation in Chapter 11 arrived last and from a different direction. Involutivity of the characteristic variety was proved analytically by Sato, Kashiwara and Kawai and algebraically by Gabber in 1981, and it explains Bernstein's inequality in terms of symplectic geometry - an explanation, not a replacement, since the elementary proof remains shorter and gives more.

The applications at the end are the most recent layer. Van den Essen's D-module proof of the stability lemma dates from the late 1980s, and Zeilberger's algorithmic programme from 1990-91. That a theory built to answer a question about distributions should be turned into a tool for proving binomial identities by machine was not anticipated by anyone in the 1970s, and it is a fair summary of why the elementary theory is worth learning in this order: the invariants come first because everything else turns out to be about them.

Comparison

The chapter map below groups the material by function rather than by number. The final column names what each group delivers to the rest of the theory.

The architecture of the elementary theory, grouped by role.
Chapters	Theme	Central results	Delivers
1-2	The Weyl algebra as a ring	canonical form; $A_n$ is a domain; $A_n$ is simple	the ground rules used everywhere
3	Rings of differential operators	the inductive definition of $D(R)$ ; $D(K[x]) = A_n$	the wider context, and derivations
4	The Jacobian conjecture	Dixmier implies Jacobian	a first application; §1 is needed again in Ch. 14
5-6	Modules and differential equations	the polynomial module; twists; the module of a system; solutions as homomorphisms	the examples every later chapter tests against
7-8	Filtered and graded modules, Noetherian theory	good filtrations; $A_n$ is Noetherian	the machinery for invariants
9	Dimension and multiplicity	the Hilbert polynomial; $d(M)$ and $e(M)$ ; Bernstein's inequality	the numerical invariants
10	Holonomic modules	finite length; cyclicity; the b-function	the good class of modules
11	Characteristic varieties	$Ch(M)$ ; involutivity; Lagrangian means holonomic	geometric meaning; used nowhere later
12-13	Tensor and external products	the universal property; $A_n+m ≅ A_n ⊗ A_m$	the algebra needed to define the operations
14-16	Inverse and direct images	$f^*$ and $f_+$ ; factoring through the graph; right modules	the operations themselves
17	Kashiwara's theorem	modules supported on a subspace are direct images from it	the structural theorem for embeddings
18	Preservation of holonomy	$f^*$ and $f_+$ preserve holonomicity	the theorem the whole book aims at
19-20	Applications	van den Essen's lemma; Zeilberger's method	evidence that the machinery pays

Read the table as three blocks: rows one to seven are the ring and its invariants, rows eight to twelve are the operations, and the last row is what they are for. The single row that is genuinely detachable is Chapter 11.

Key Takeaways

Key points

The theory has two halves: invariants of $A_n$ -modules, then operations induced by polynomial maps.
Chapters on the ring itself - canonical form, domain, simple, Noetherian - are used everywhere and cannot be skipped.
Filtrations reduce noncommutative questions to commutative Hilbert-function theory; this is the engine of the invariant theory.
Bernstein's inequality confines $d(M)$ to $[n,2n]$ and defines the holonomic class, which carries all the finiteness.
The operations are built from tensor products by factoring a map into an embedding and a projection.
The central theorem is that inverse and direct images preserve holonomicity - even though they do not preserve dimension.
Characteristic varieties are the one genuinely optional chapter: they depend on everything and are used by nothing later.
The two applications, dynamical stability and automatic identity proving, are independent of each other and of most of the operations.

FAQs

What is the shortest route to understanding holonomic modules?

Canonical form and simplicity of $A_n$ ; the polynomial module as the standard example; filtered and graded modules; good filtrations; the Hilbert polynomial; Bernstein's inequality; then the definition. That is roughly seven chapters' worth of material and none of it requires geometry. The operations and the characteristic variety can both be deferred.

Do I need the tensor product chapter if I already know tensor products?

Probably not for the construction, but check two things first: the treatment of bimodules, since the image functors depend on a module being simultaneously a left module over one ring and a right module over another, and the identification of localisation as a tensor product, which is how the inverse image is computed in practice.

Why are there separate chapters for embeddings and for general maps?

Because an arbitrary polynomial map is factored as a closed embedding - into its graph - followed by a projection, and the two pieces behave completely differently. The embedding case is rigid and is governed by Kashiwara's theorem; the projection case is where the integration happens. Handling them separately is what makes the general construction manageable.

Is Chapter 11 really optional?

Logically, within the elementary theory, yes: nothing after it cites it. Mathematically it is where the subject acquires its geometric meaning, and every advanced treatment starts from the characteristic variety rather than from the Hilbert polynomial. Skipping it is a reasonable choice for a first pass and a poor one for a second.

How much algebraic geometry do I actually need?

For everything except the characteristic variety chapter, none. For that chapter: affine varieties, the correspondence between radical ideals and closed sets, irreducible components, and the dimension of an affine variety. A first chapter of a standard textbook covers it. Symplectic linear algebra is developed from scratch where it is needed.

Where does the theory become genuinely hard?

At two places. The proof that dimension is independent of the good filtration is the first genuinely technical argument. The preservation theorems for the image functors are the second, and they are hard for a different reason: the constructions involve several successive tensor products and the bookkeeping is heavy. Everything before Chapter 9 is elementary in the strict sense.

Can I learn the algorithmic side without the module theory?

Up to a point. Creative telescoping can be presented as a manipulation of recurrences and certificates, and many practitioners use it that way. What the module theory supplies is the guarantee that the algorithm terminates and that the class of inputs is closed under the operations performed - and those are exactly the questions a user hits when the algorithm fails to return.

What comes after this collection?

Sheaves of differential operators on smooth varieties, the derived category formalism for the six operations, regular holonomicity, and the Riemann-Hilbert correspondence; then perverse sheaves and geometric representation theory. The vocabulary transfers directly. What has to be learned is the homological and sheaf-theoretic machinery that the affine setting made unnecessary.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Introduction §3-§5, which sets out the plan of the book and its prerequisites.
I. N. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Functional Analysis and its Applications 6 (1972), 273-285 - the origin of the invariant theory.
M. Kashiwara, Algebraic study of systems of partial differential equations, thesis, Tokyo, 1970; translated in Mémoires de la Société Mathématique de France 63 (1995) - the origin of the operations.
J.-E. Björk, Rings of Differential Operators, North-Holland Mathematical Library 21, North-Holland, 1979 - the standard reference for the filtered theory.
R. Hotta, K. Takeuchi and T. Tanisaki, D-modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics 236, Birkhäuser, 2008 - the continuation of this roadmap into the general theory.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, revised edition, American Mathematical Society, 2001 - for the ring-theoretic background used in Chs. 7-8.
M. Petkovšek, H. S. Wilf and D. Zeilberger, A = B, A. K. Peters, 1996 - the algorithmic application of Ch. 20.
S. Saito, B. Sturmfels and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6, Springer, 2000 - the computational counterpart to the invariant theory.
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization.

AI Suggested Questions

Draw the dependency graph of the twenty chapters and identify every vertex with no outgoing edge.
Compute $d(M)$ and $e(M)$ for $M = A_1/A_1(x∂ - 3)$ and locate it on the roadmap.
Explain why the direct image of a holonomic module can have a different dimension from the original.
Give an example of a finitely generated $A_2$ -module with $d(M) = 3$ , strictly between $2$ and $4$ .
Describe what changes in the roadmap if the base is a smooth affine variety instead of affine space.
Trace the module $K[x][1/x]$ through the same five stages as the worked example on this page.
Identify which milestone theorems are used in the proof that holonomic modules have finite length.