Systems of Differential Equations and Presentation Matrices

Q: Can every system be reduced to one equation in one unknown?

Not over A_n : the free module A_n^2 has rank two over the quotient division ring of A_n and a cyclic module has rank at most one. Over B_1 , with rational function coefficients, the cyclic vector theorem says yes for systems of the usual kind, and over A_n it holds for holonomic modules, though not by any cheap algorithm.

Q: What exactly are the compatibility conditions?

The syzygies: row vectors Q with QP = 0 . Applied to an inhomogeneous system Pf = g they give the necessary conditions Qg = 0 . For the gradient system in two variables the single generating syzygy reproduces the classical condition ∂_2 g_1 = ∂_1 g_2 .

Q: Is the Cauchy-Riemann system holonomic?

No, and it must not be. Over a field containing i the system decouples into the two single equations (∂_1 + i∂_2)w = 0 and (∂_1 - i∂_2) w = 0 , so its module is a direct sum of two modules of the form A_2/A_2P with P of Bernstein degree one. Each summand has dimension 2n-1 = 3 , so d(M) = 3 > 2 and e(M) = 2 . The infinite-dimensional space of holomorphic functions on an open set is exactly what a non-holonomic module of this kind should have.

Overview

One unknown function and finitely many equations gave a cyclic module $A_n/J$ . Most systems that arise in practice have several unknown functions: the Cauchy-Riemann equations, a first-order system obtained by reducing the order of a scalar equation, a connection written in a local frame. The construction extends without difficulty, and the object that appears is a quotient of a free module rather than of the ring.

Concretely, a system of $m$ equations in $k$ unknowns is a matrix $P = (P_ij)$ of operators, and the module of the system is the cokernel of the map of free left modules $A_n^m → A_n^k$ given by right multiplication by $P$ . Solutions in a target module $S$ are the columns $s ∈ S^k$ with $Ps = 0$ , and these are again exactly the homomorphisms out of the cokernel, by the same argument as in the cyclic case.

Two facts make the extension worth stating separately. First, it is not a genuine enlargement of the class of modules involved: because $A_n$ is Noetherian, every finitely generated $A_n$ -module is finitely presented, so every such module is the module of some system - and in the holonomic case it can even be presented with one unknown. Second, the matrix is far from unique. Row operations, column operations, and adding trivial equations all leave the module unchanged, so any invariant read off the matrix must be checked against those moves.

The rows of $P$ that are consequences of the others, and the relations among the rows - the syzygies - carry real information: they are exactly the compatibility conditions that an inhomogeneous right-hand side must satisfy.

Definition

Fix the convention that $A_n^k$ is the free left $A_n$ -module of rank $k$ with basis $e_1, … ,e_k$ , written as row vectors, and that unknown functions are gathered into a column.

The module of a system in several unknowns

Let $P = (P_ij) ∈ M_m × k(A_n)$ and consider the system in unknowns $f_1, … ,f_k$

∑ _j=1^k P_ij, f_j = 0 (i = 1, … ,m).

(6.14)

Let $r_i = ∑ _j P_ij e_j ∈ A_n^k$ be the $i$ -th row of $P$ and let $N = ∑ _i A_n r_i$ be the left submodule they generate. The module of the system is

M = A_n^k / N = coker (A_n^m ⟶ ⋅ ,P A_n^k ),

(6.15)

where the map sends a row vector $Q = (Q_1, … ,Q_m)$ to the matrix product $Q P$ . Right multiplication by a matrix is a homomorphism of left modules, which is why the matrix multiplies on that side.

Solutions of a matrix system

For every left $A_n$ -module $S$ there is an isomorphism of $K$ -vector spaces

Hom_A_n(M, S) ; ≅ ; , s = (s_1, … ,s_k)^T ∈ S^k : P s = 0 ,, φ ↦ (φ( e_1), … ,φ( e_k))^T,

where $e_j$ is the class of $e_j$ in $M$ . In words: the $j$ -th unknown function is the image of the $j$ -th basis vector.

Proof

A homomorphism $A_n^k → S$ is determined freely by the images $s_j$ of the basis vectors, giving $Hom_A_n(A_n^k,S) ≅ S^k$ . Such a homomorphism factors through $M$ exactly when it kills every $r_i$ , and $r_i ↦ ∑ _j P_ijs_j$ , which is the $i$ -th entry of $P s$ . So factoring through $M$ is precisely the system (6.14) holding for $(s_1, … ,s_k)$ .

Everything finitely generated is a system

If $M$ is any finitely generated left $A_n$ -module, choose generators to get a surjection $A_n^k ↠ M$ ; its kernel is finitely generated because $A_n$ is left Noetherian, so a choice of generators of the kernel produces a matrix $P$ with $M ≅ coker( ⋅ P)$ . Hence finitely generated modules and systems of linear differential equations with polynomial coefficients are the same subject, viewed twice.

Core Concepts

Rows are equations, columns are unknowns

The shape of the matrix is the shape of the problem: $m$ rows, one per equation; $k$ columns, one per unknown. The free module $A_n^k$ is the system with no equations at all, whose solution space is all of $S^k$ . Imposing equations means dividing out the submodule they generate, and the solution space shrinks accordingly.

The module is the invariant, the matrix is not

Three families of moves change $P$ without changing $M$ up to isomorphism. Replacing $P$ by $U P$ for an invertible $U ∈ GL_m(A_n)$ replaces the generators of $N$ by another generating set. Replacing $P$ by $P V$ for an invertible $V ∈ GL_k(A_n)$ is a change of basis of $A_n^k$ , that is, a linear change of unknown functions with operator coefficients. Adding a row of zeros, or deleting a row that is a left combination of the others, changes nothing at all.

There is a fourth, less obvious move: adding a redundant unknown together with the equation defining it. If $P$ is enlarged to a block matrix with an extra unknown $f_k+1$ and the extra equation $f_k+1 = 0$ , the cokernel is unchanged. This is how a system of the wrong size is brought into a normal form, and it is why the pair $(m,k)$ carries no invariant meaning.

Syzygies are compatibility conditions

A syzygy of $P$ is a row vector $Q ∈ A_n^m$ with $QP = 0$ . Applying $Q$ to the inhomogeneous system $Pf = g$ gives $0 = Qg$ , so every syzygy imposes a condition on the right-hand side. For the classical pair of equations $∂_1 f = g_1$ , $∂_2 f = g_2$ in one unknown, the syzygy $(∂_2, -∂_1)$ of the matrix $P = (∂_1, ∂_2)^T$ yields the familiar necessary condition $∂_2 g_1 = ∂_1 g_2$ . Continuing to take syzygies of syzygies produces a free resolution of $M$ , and the whole resolution is the correct home for the theory of over-determined systems.

Key Equations

The presentation and the solution correspondence, side by side:

A_n^m ⟶ ⋅ ,P A_n^k ⟶ M ⟶ 0, Hom_A_n(M,S) ≅ (P : S^k → S^m ).

(6.16)

A single equation of order $k$ becomes a first-order system through its companion matrix. For $P = ∂^k + a_k-1∂^k-1 + … + a_0$ with $a_i ∈ K[x]$ , putting $f_j = ∂^,j-1f$ gives

∂ pmatrix f_1 f_2 ⋮ f_k pmatrix = pmatrix 0 & 1 & & & ⋱ & ⋱ & & & 0 & 1 -a_0 & -a_1 & ⋅ s & -a_k-1 pmatrix pmatrix f_1 f_2 ⋮ f_k pmatrix,

(6.17)

and the module of the system on the right is isomorphic to $A_1/A_1P$ . Conversely, over the ring $B_1 = K(x) ⟨ ∂ ⟩$ of operators with rational function coefficients, every first-order system of size $k$ with invertible leading matrix is equivalent to a single scalar equation of order $k$ :

coker (B_1^k ⟶ ⋅ ,(∂I - Λ) B_1^k ) ; ≅ ; B_1/B_1L, ordL = k,

(6.18)

the cyclic vector theorem; $Λ$ is a $k \times k$ matrix over $K(x)$ and $L$ the resulting scalar operator.

For the dimension theory, a presentation gives an upper bound directly: $M$ is a quotient of $A_n^k$ , so

d(M) ≤ d(A_n^k) = 2n, e(A_n^k) = k,

(6.19)

and by Bernstein's inequality $d(M) ≥ n$ unless $M = 0$ .

Variable Definitions

$m$: the number of equations, hence the number of rows
$k$: the number of unknown functions, hence the number of columns
$P$: the presentation matrix, an $m \times k$ matrix over $A_n$
$e_1, … ,e_k$: the standard basis of the free module $A_n^k$
$r_i$: the $i$ -th row of $P$ , read as an element of $A_n^k$
$N$: the submodule $∑ _i A_n r_i$ of relations
$M$: the module of the system, $A_n^k/N$
$e_j$: the class of $e_j$ in $M$ , corresponding to the $j$ -th unknown function
$Q$: a syzygy: a row vector in $A_n^m$ with $QP = 0$
$B_1$: the ring $K(x) ⟨ ∂ ⟩$ of operators with rational function coefficients

Properties and Behaviour

Presentations always exist and are never unique

Every finitely generated left $A_n$ -module has a finite presentation, and two matrices $P$ , $P^{'}$ present isomorphic modules if and only if they can be connected by a finite sequence of the moves listed above together with stabilisation by identity blocks. Nothing forces a minimal presentation to be unique.

Rank obstruction to a single unknown

$A_n$ is a Noetherian domain, hence an Ore domain with a quotient division ring $D$ . For a finitely generated $M$ the rank $_D (D ⊗ _A_n M)$ is well defined and additive, and a cyclic module has rank at most $1$ . Therefore $A_n^k$ for $k ≥ 2$ is not cyclic: no system of $k ≥ 2$ unknowns and no equations can be rewritten with a single unknown.

But holonomic systems canCoutinho, Ch. 10

Every holonomic $A_n$ -module is cyclic. So whenever the system is holonomic - which is the case for a first-order system in one variable, or for a maximally over-determined system in several - there exists a single unknown and a single left ideal presenting the same module. The proof is not constructive in a useful sense, and the resulting scalar equation may be enormous; over $B_1$ the cyclic vector theorem gives an effective version.

The count of equations decides nothing

A system with more equations than unknowns can have a large solution space, and one with fewer can have none. Two equations in one unknown, $∂_1 f = ∂_2 f = 0$ in $A_2$ , present $K[x_1,x_2]$ , with a one-dimensional space of polynomial solutions. One equation in one unknown, $xf = 0$ in $A_1$ , has no non-zero solution in any classical target. Only the module decides.

Generation by two elements

Stafford proved that every finitely generated left $A_n$ -module can be generated by two elements, so $k ≤ 2$ can always be arranged in principle. This is a theorem about the Weyl algebra specifically and is much stronger than what holds for a general Noetherian ring; it does not make the resulting presentation easy to find.

Examples and Special Cases

The Cauchy-Riemann system

Two unknowns $u, v$ and two equations $∂_1 u - ∂_2 v = 0$ , $∂_2 u + ∂_1 v = 0$ , over $A_2$ . The presentation matrix is

P = pmatrix ∂_1 & -∂_2 ∂_2 & ∂_1 pmatrix.

Over a field containing $i$ the system decouples. Writing $J$ for the matrix of the rotation by a quarter turn, $P = ∂_1 I + ∂_2 J$ with $J^2 = -I$ , so conjugating by the matrix of eigenvectors of $J$ turns $P$ into $diag(∂_1 + i∂_2,; ∂_1 - i∂_2)$ . Hence $M ≅ A_2/A_2(∂_1+i∂_2) ; ⊕ ; A_2/A_2(∂_1-i∂_2)$ , and by the single-equation formula on the companion page each summand has dimension $2n-1 = 3$ and multiplicity $1$ : the system is not holonomic, which is as it should be, since holomorphic functions on an open set form an infinite-dimensional solution space.

Left combinations of the rows produce consequences. Taking $∂_1 r_1 + ∂_2 r_2$ gives $((∂_1^2 + ∂_2^2),, -∂_1∂_2 + ∂_2∂_1) = (Δ, 0)$ , and $-∂_2 r_1 + ∂_1 r_2$ gives $(0, Δ)$ , where $Δ = ∂_1^2 + ∂_2^2$ . So $Δ e_1 = Δ e_2 = 0$ in $M$ : the module knows that the real and imaginary parts of a holomorphic function are harmonic, and the computation is two lines of operator algebra with no analysis in it.

No equations

$P$ the empty matrix, $M = A_n^k$ . Then $Hom_A_n(M,S) = S^k$ : every $k$ -tuple is a solution, as it should be. This is the case where the module is free and the solution space is as large as the target allows.

An inconsistent system that does not look inconsistent

One unknown, two equations $∂ f = 0$ and $(∂ + 1)f = 0$ over $A_1$ . Subtracting the rows gives $1 ∈ J$ , so $M = 0$ and the only solution in any target is $0$ . In matrix form, $P = (∂, ∂+1)^T$ has the syzygy $(1,-1)$ and the left submodule generated by its rows is all of $A_1$ .

Compatibility for the gradient system

One unknown, $P = (∂_1, ∂_2)^T$ over $A_2$ , so $M = A_2/(A_2∂_1 + A_2∂_2) ≅ K[x_1,x_2]$ . The syzygy module of $P$ is generated by $(∂_2, -∂_1)$ , and the inhomogeneous system $∂_1 f = g_1$ , $∂_2 f = g_2$ therefore requires $∂_2 g_1 - ∂_1 g_2 = 0$ . Over $K[X]$ that condition is also sufficient, which is the polynomial Poincaré lemma.

Worked Example

The Airy equation as a first-order system

Step 1 - the two descriptions
Scalar form: $f'' = xf$ , that is $Pf = 0$ with $P = ∂^2 - x ∈ A_1$ , giving the cyclic module $M_1 = A_1/A_1P$ .

System form: set $f_1 = f$ and $f_2 = f'$ , so that $f_1' - f_2 = 0$ and $f_2' - xf_1 = 0$ . The presentation matrix is

$Λ = pmatrix ∂ & -1 -x & ∂ pmatrix, M_2 = A_1^2 / (A_1 r_1 + A_1 r_2 ),$

with rows $r_1 = ∂ e_1 - e_2$ and $r_2 = -x e_1 + ∂ e_2$ .
Step 2 - a map from the system module to the scalar module
Define $β : A_1^2 → A_1/A_1P$ on the basis by $β(e_1) = 1$ and $β(e_2) = ∂$ , extending $A_1$ -linearly. Check that the rows die:

$β(r_1) = ∂ 1 - ∂ = 0, β(r_2) = -x 1 + ∂, ∂ = ∂^2 - x = 0.$

So $β$ factors through a homomorphism $β : M_2 → M_1$ .
Step 3 - a map back
Define $γ : A_1 → M_2$ by $γ(Q) = Q e_1$ . Its kernel contains $P$ , because in $M_2$ the first row gives $e_2 = ∂ e_1$ and the second then gives $∂^2 e_1 = ∂ e_2 = x e_1$ , so $(∂^2 - x) e_1 = 0$ . Hence $γ$ factors through $γ : M_1 → M_2$ with $γ( 1) = e_1$ .
Step 4 - the two maps are mutually inverse
$β γ( 1) = β( e_1) = 1$ , so $\bar{β} \bar{γ}$ is the identity on the generator of $M_1$ , hence on $M_1$ . In the other direction, $γ β( e_1) = e_1$ and $γ β( e_2) = γ( ∂) = ∂ e_1 = e_2$ , using the first-row relation. Since $e_1, e_2$ generate $M_2$ , the composite is the identity. Therefore $M_2 ≅ M_1$ .
Step 5 - invariants and solutions, computed once
The Bernstein degree of $∂^2 - x$ is $2$ - the term $∂^2$ has degree $2$ and $x$ has degree $1$ - and its symbol is $ξ^2$ . So $grM_1 ≅ K[x,ξ]/(ξ^2)$ , whose Hilbert function in degree at most $m$ counts the monomials $x^a$ and $x^aξ$ : that is $(m+1) + m = 2m+1$ . Hence $d(M_1) = 1$ and $e(M_1) = 2$ , so the module is holonomic, and the same holds for $M_2$ .

Check the count directly at $m = 2$ : $B_2$ has the six monomials $1,x,∂,x^2,x∂,∂^2$ , and $B_2 ∩ A_1P = KP$ , so $Γ_2 = 5 = 2 ⋅ 2 + 1$ .

On solutions, the isomorphism reads: pairs $(f_1,f_2) ∈ S^2$ with $f_1' = f_2$ , $f_2' = xf_1$ correspond to single functions $f ∈ S$ with $f'' = xf$ , by $f = f_1$ and $f_2 = f'$ . Over $S = K[x]$ both spaces are zero, since a non-zero polynomial solution of $f'' = xf$ would have $(f'') = f - 2$ on the left and $f + 1$ on the right.

Result

The companion system $Λ$ and the scalar operator $∂^2 - x$ present isomorphic $A_1$ -modules, with $d = 1$ and $e = 2$ ; the module is holonomic. Order reduction is an isomorphism of modules, not merely a bijection of solution sets, so every module-theoretic invariant of the Airy equation may be computed from whichever presentation is more convenient. Neither has a non-zero polynomial solution.

Applications and Industry Use

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

Symbolic solvers. Computer algebra systems move between a scalar equation and a first-order system routinely; doing so is legitimate precisely because the two present isomorphic modules, and every invariant used to classify singularities transfers.
Multidimensional control. In the module-theoretic approach to linear systems, a controlled plant is a presentation matrix over a ring of operators and structural properties such as controllability, observability and autonomy are properties of the cokernel. Free cokernels correspond to flat systems, and testing freeness is a computation on the matrix.
Integrable connections. A flat connection on a trivial bundle of rank $k$ is a system $∂_i f = A_i f$ ; the associated module is the one presented by the matrices $∂_i I - A_i$ , and flatness is exactly the condition that the syzygies close up.
Hypergeometric systems. The GKZ systems attached to a lattice are given as explicit ideals in $A_n$ with several unknowns hidden in a torus action; their holonomic rank is computed from the presentation and equals a volume in the generic case.
Over-determined systems in physics. Maxwell-type systems and the Cauchy-Riemann example above are over-determined, and their compatibility conditions - the syzygies - are the familiar identities such as the vanishing of a divergence or a curl that any admissible source term must satisfy.

Design Considerations

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

Choosing the number of unknowns

Scalar presentations are convenient for theory: one generator, one left ideal, invariants easy to define. Matrix presentations are convenient for computation: the entries stay of low order, whereas eliminating unknowns to obtain a scalar equation inflates both order and coefficient size. The cyclic vector theorem guarantees the scalar form exists over $B_1$ , but the coefficient growth it causes is the reason numerical and symbolic implementations usually keep the system form.

Choosing the ring

Over $A_n$ the coefficients are polynomials and the singular locus is visible; over $B_1$ the coefficients are rational functions, the ring is a principal ideal domain, and matrices can be reduced to a Jacobson normal form, at the price of erasing the singularities. Deciding which questions must survive the change is the main design decision when a system is put into normal form.

Row echelon thinking does not transfer

Over a field one reduces a matrix to echelon form and reads everything off. Over $A_n$ there is no such normal form: the ring is neither commutative nor a principal ideal domain, and Gröbner bases for submodules of $A_n^k$ are the substitute. They give normal forms for elements, not for matrices.

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.

Fix a monomial order on $A_n^k$ that refines a filtration and is compatible with the module structure; the standard choices are position-over-term and term-over-position.
Compute a Gröbner basis of the submodule $N$ generated by the rows. Membership in $N$ , and hence whether a candidate equation is a consequence of the system, becomes reduction to zero.
Compute the syzygy module of the rows in the same computation; its generators are the compatibility conditions for the inhomogeneous problem.
Iterate to obtain a free resolution of $M$ ; over $A_n$ in characteristic zero the global dimension is $n$ , so a free resolution of length at most $n$ always exists.
Read the invariants from the leading term module: the Hilbert polynomial of $gr M$ gives $d(M)$ and $e(M)$ , and the holonomic rank after extending coefficients to rational functions gives the expected number of independent local solutions.

Implementations: Dmodules in Macaulay2, dmod.lib and ncalg.lib in Singular, ore_algebra in SageMath, and the OreModules package built on Maple, which is written specifically around presentation matrices and system-theoretic properties of their cokernels. All of these represent a system as a matrix and a module as a matrix; the isomorphism class is never stored, only recomputed.

Limits of Validity

No Smith normal form over $A_n$ . Diagonal reduction of a matrix of operators is available over $B_1$ and over other principal ideal domains, not over $A_n$ itself, and not in several variables. Algorithms that assume it will fail on systems with singularities.
The pair $(m,k)$ is not an invariant. Adding trivial equations or redundant unknowns changes it freely. Any statement of the form the system has more equations than unknowns, therefore ... is a statement about a presentation, not about the system.
Cyclic presentations need not be effective. Every holonomic module is cyclic, but finding the cyclic generator and the resulting scalar operator can be prohibitively expensive, and the resulting operator can have very large coefficients.
Inhomogeneous solvability needs more than syzygies. Applying every syzygy to the right-hand side gives necessary conditions. Their sufficiency is a statement about the vanishing of an $Ext$ group in the chosen target and depends on the target, not only on the matrix.

Failure Modes and Common Mistakes

Multiplying the matrix on the wrong side

For left modules, a homomorphism $A_n^m → A_n^k$ is right multiplication by a matrix, and consequences of the equations are obtained by multiplying rows on the left by operators. Writing $P f$ with $f$ a column of functions and $Q P$ with $Q$ a row of operators keeps this straight. Reversing either convention produces expressions that are not $A_n$ -linear, and in a noncommutative ring that error does not cancel.

Treating a first-order system as a different problem from a scalar equation

They present isomorphic modules whenever the reduction of order is performed correctly, as the Airy example shows. Computing an invariant from the system form and comparing it with the same invariant computed from the scalar form is a useful consistency check, and any discrepancy indicates an error in the reduction rather than a real difference.

Reading the order of the matrix entries as the rank of the solution space

The entries of $Λ$ in the Airy example have order one, yet the local solution space has dimension two. The invariant that counts solutions is the holonomic rank of the module, computed after extending the coefficients to rational functions; it is not the maximum order of an entry, nor the size of the matrix.

Assuming redundant rows are harmless to detect

Deciding whether a row is a consequence of the others is a Gröbner basis computation in $A_n$ , not an inspection. Rows can be consequences in ways that involve high-order operators - the harmonicity of $e_1$ in the Cauchy-Riemann example is a second-order consequence of first-order equations - so removing a row because it looks independent is unsafe.

Historical Notes

The formal theory of over-determined systems was built by Riquier and Janet in the first decades of the twentieth century, with the notion of a passive or involutive system - one closed under its own integrability conditions - as the central object; Janet bases are a direct ancestor of Gröbner bases and were introduced for exactly the completion problem described above. Cartan and Kähler gave the differential-geometric counterpart, and Spencer recast the theory cohomologically in the 1960s.

Presenting the system as a matrix over a ring of operators, and studying the cokernel, is the algebraic distillation of that programme. Jacobson's normal form for matrices over noncommutative principal ideal domains dates from the 1930s and 1940s and underlies the cyclic vector theorem; the systematic use of the cokernel as the system is due to Ehrenpreis and Palamodov in the constant-coefficient case and to Kashiwara in general. In control theory the same move was made independently in the 1990s by Oberst, Fliess and Pommaret.

Comparison

One unknown versus several.
	Single unknown	Several unknowns
Data	operators $P_1, … ,P_m ∈ A_n$	matrix $P ∈ M_m × k(A_n)$
Relations submodule	left ideal $J ⊂ eq A_n$	submodule $N ⊂ eq A_n^k$
Module	$A_n/J$ , cyclic	$A_n^k/N$ , finitely presented
Solutions in $S$	$s ∈ S : Js = 0$	$(P : S^k → S^m)$
Always achievable?	only if the module is cyclic	always
Guaranteed when	the module is holonomic	the module is finitely generated
Typical cost of conversion	coefficient blow-up	none

Key Takeaways

Key points

A system of $m$ equations in $k$ unknowns is a matrix $P$ over $A_n$ , and its module is $coker(A_n^m ⟶ ⋅ ,P A_n^k)$ .
Solutions in a target $S$ are the columns $s ∈ S^k$ with $Ps = 0$ , and these are exactly the homomorphisms from the cokernel into $S$ .
Because $A_n$ is Noetherian, every finitely generated $A_n$ -module is finitely presented: modules and systems are the same subject.
The matrix is not an invariant. Row operations, column operations, zero rows and redundant unknowns all leave the module unchanged.
Syzygies of the matrix are the compatibility conditions on the right-hand side of an inhomogeneous system; iterating them builds a free resolution.
Reducing a scalar equation of order $k$ to a first-order system is an isomorphism of modules; the converse, a cyclic vector, exists over rational function coefficients and for holonomic modules, but not in general.

FAQs

Why does the matrix multiply on the right?

Because the modules are left modules. Right multiplication by a fixed matrix commutes with left multiplication by ring elements, so it is a homomorphism of left modules; left multiplication by a matrix is not, unless the ring is commutative.

Can every system be reduced to one equation in one unknown?

Not over $A_n$ : the free module $A_n^2$ has rank two over the quotient division ring of $A_n$ and a cyclic module has rank at most one. Over $B_1$ , with rational function coefficients, the cyclic vector theorem says yes for systems of the usual kind, and over $A_n$ it holds for holonomic modules, though not by any cheap algorithm.

Does an over-determined system have fewer solutions?

Not necessarily, because equations can be redundant. What is true is that adding a genuine equation shrinks the module and hence the solution space in every target simultaneously. Whether an added equation is genuine is a Gröbner basis question.

What exactly are the compatibility conditions?

The syzygies: row vectors $Q$ with $QP = 0$ . Applied to an inhomogeneous system $Pf = g$ they give the necessary conditions $Qg = 0$ . For the gradient system in two variables the single generating syzygy reproduces the classical condition $∂_2 g_1 = ∂_1 g_2$ .

How do I know whether two matrices present the same module?

By computing rather than by inspection: build the two cokernels, attempt to construct homomorphisms both ways on generators as in the Airy example, and verify that the composites are the identity on generators. There is no normal form to compare against in general.

Is the number of independent solutions the size of the matrix?

No. It is the holonomic rank of the module, which for a first-order system of size $k$ with invertible leading matrix equals $k$ , but for a general presentation has no relation to $m$ or $k$ .

Do the invariants

d(M)

and

e(M)

depend on the presentation?

No, they depend only on the isomorphism class of the module, since they are defined through any good filtration. The presentation only affects how they are computed, and a presentation with $k$ generators gives the crude bound $e(M) ≤ k ⋅ e(A_n)$ before any relations are used.

Is the Cauchy-Riemann system holonomic?

No, and it must not be. Over a field containing $i$ the system decouples into the two single equations $(∂_1 + i∂_2)w = 0$ and $(∂_1 - i∂_2) w = 0$ , so its module is a direct sum of two modules of the form $A_2/A_2P$ with $P$ of Bernstein degree one. Each summand has dimension $2n-1 = 3$ , so $d(M) = 3 > 2$ and $e(M) = 2$ . The infinite-dimensional space of holomorphic functions on an open set is exactly what a non-holonomic module of this kind should have.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 6 §1 for the cyclic case, Ch. 8 for the Noetherian property, Ch. 10 for cyclicity of holonomic modules.
M. Kashiwara, Algebraic Study of Systems of Partial Differential Equations, master's thesis, University of Tokyo 1970; Mémoires de la Société Mathématique de France 63, 1995.
N. Jacobson, The Theory of Rings, Mathematical Surveys 2, American Mathematical Society, 1943 - normal forms for matrices over noncommutative principal ideal domains.
J. T. Stafford, Module structure of Weyl algebras, Journal of the London Mathematical Society 18 (1978), 429-442 - every finitely generated module needs at most two generators.
U. Oberst, Multidimensional constant linear systems, Acta Applicandae Mathematicae 20 (1990), 1-175, and J.-F. Pommaret, Partial Differential Control Theory, Kluwer, 2001 - presentation matrices in systems theory.
M. Saito, B. Sturmfels and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Springer, 2000 - Gröbner bases in free modules over the Weyl algebra and holonomic rank.
F. Chyzak, A. Quadrat and D. Robertz, OreModules: a symbolic package for the study of multidimensional linear systems, in Applications of Time Delay Systems, Springer, 2007.
Macaulay2 Dmodules package and Singular dmod.lib documentation, for computations with matrices over the Weyl algebra.

AI Suggested Questions

Reduce the equation $f''' = x f$ to a first-order system and prove the two modules are isomorphic.
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