Polynomial Maps and the Jacobian Matrix

Q: Why is the comorphism written with the arrow reversed?

Because functions pull back. If g is a function on the target Y and F maps X into Y , then g ∘ F is a function on X . There is no way to push a function forward along a non-injective map without extra choices, so the natural operation goes from K[Y] to K[X] .

Q: Does Δ F = 1 imply F is injective?

It implies that the comorphism F^# is injective, equivalently that F_1, … ,F_n are algebraically independent; that is proved on the conjecture page and needs only Δ F ≠ 0 everywhere. Injectivity of F itself as a map of points is known over C only in special cases, for instance when all coordinate functions have degree at most 2 .

Q: If the Jacobian never vanishes, does the inverse function theorem not finish the job?

No. It gives a local inverse near each point, defined only on a neighbourhood and only analytic. Two obstacles remain: the local inverses must agree on overlaps and cover the whole space, which is a global topological question, and the resulting global inverse must be polynomial, which the theorem says nothing about. The example x ↦ x^3 + x over R clears the first hurdle and fails the second.

Q: Is the Jacobian conjecture just the statement that Δ F is a unit?

The other way round. That Δ F is a unit is the easy, proved direction. The conjecture is the converse: that Δ F ∈ K^ × forces the existence of a polynomial inverse.

Q: What changes in positive characteristic?

Two things break. Polynomials are no longer determined by their values on K^n when K is finite, and the differential calculus loses information because p -th powers have zero derivative. The map F(x_1, … ,x_n) = (x_1 + x_1^,p, x_2, … , x_n) over a field of characteristic p has Jacobian matrix the identity yet is not invertible, since K[x_1 + x_1^,p] contains no element of degree 1 . See the positive characteristic page for the parallel failure on the operator side.

Q: How do I actually check whether a given map is invertible?

Compute Δ F first; if it is not a non-zero constant you are done, the map is not invertible. If it is, try to express each x_j as a polynomial in F_1, … ,F_n , either by hand as in the worked example or by an elimination Gröbner basis. Success proves invertibility; failure proves nothing, because the search may simply have been out of reach.

Q: Why do polynomial maps matter for D-modules at all?

Because the functors that make the theory useful are attached to maps. The inverse image of a D-module along F is built from the comorphism F^# together with the chain rule, which is precisely the Jacobian matrix telling you how derivations transform. The material on this page is the input to that construction.

Overview

Almost every geometric construction in this collection eventually gets translated into algebra, and the translation device is always the same: a map between affine spaces is replaced by a homomorphism between rings of functions, running the other way. This page sets up that dictionary in the simplest case, where the spaces are affine spaces $K^n$ and $K^m$ and the rings are polynomial rings.

Two things come out of it. The first is a clean criterion: a polynomial map is an isomorphism exactly when the induced homomorphism of polynomial rings is an isomorphism. That turns a question about the existence of an inverse map into a question about generation of a ring, and the second form is often the tractable one.

The second is the Jacobian matrix. For a polynomial map $F : K^n → K^n$ the chain rule forces the Jacobian determinant $Δ F$ to be a unit of $K[x_1, … ,x_n]$ whenever $F$ is invertible, and the units of a polynomial ring over a field are just the non-zero scalars. So $Δ F ∈ K^ ×$ is a necessary condition for invertibility, and it is a condition one can check mechanically. Whether it is also sufficient is the Jacobian conjecture, open since 1939.

The same dictionary is used again much later, when polynomial maps are used to move D-modules around: the ring map induced by a polynomial map is exactly the comorphism defined here, and it is what makes the inverse image functor possible.

Definition

Throughout, $K$ is a field of characteristic zero, $X = K^n$ with coordinate ring $K[X] = K[x_1, … ,x_n]$ , and $Y = K^m$ with coordinate ring $K[Y] = K[y_1, … ,y_m]$ .

Polynomial mapCoutinho, Ch. 4 §1

A map $F : X \to Y$ is polynomial if there exist polynomials $F_1, … ,F_m ∈ K[X]$ such that

F(p) = ( F_1(p), … , F_m(p) ) for every p ∈ X.

The $F_i$ are the coordinate functions of $F$ . A polynomial map is a polynomial isomorphism if there is a polynomial map $G : Y \to X$ with $G ∘ F = id_X$ and $F ∘ G = id_Y$ .

ComorphismCoutinho, Ch. 4 §1

Let $F : X \to Y$ be a polynomial map. Its comorphism is the map

F^# : K[Y] ⟶ K[X], F^#(g) = g ∘ F.

It is a homomorphism of $K$ -algebras, and it is determined by its values on the variables: $F^#(y_i) = F_i$ .

Jacobian matrix and Jacobian determinant

The Jacobian matrix of $F$ is the $m \times n$ matrix $J(F)$ over $K[X]$ with entries

J(F)_ij = (∂ F_i)/(∂ x_j), 1 ≤ i ≤ m, ; 1 ≤ j ≤ n.

When $m = n$ the Jacobian determinant is the polynomial $Δ F = J(F) ∈ K[X]$ . Its value at a point $p$ is written $Δ F(p)$ .

Core Concepts

Polynomials and the functions they induce

The whole construction rests on being allowed to confuse a polynomial with the function it defines. A polynomial $g ∈ K[Y]$ gives a function $Y \to K$ , $p ↦ g(p)$ . If $g = 0$ then that function is zero; the point that matters is the converse. Over an infinite field a polynomial that vanishes at every point of $K^m$ is the zero polynomial, proved by induction on $m$ : writing $g = ∑ _i g_i(y_1, … ,y_m-1) y_m^,i$ and freezing the first $m-1$ variables at a point where some $g_i$ is non-zero produces a non-zero one-variable polynomial with infinitely many roots.

Because a field of characteristic zero contains $ℚ$ and is therefore infinite, the identification is legitimate in our setting. It is what lets us define $F^#(g)$ as a polynomial rather than merely as a function, and it is the reason arguments in this chapter can move freely between "holds at every point of $K^n$ " and "holds as an identity of polynomials".

The arrow turns around

Composing with $F$ pulls functions back: a function on the target becomes a function on the source. So $F : X \to Y$ produces $F^# : K[Y] → K[X]$ . This reversal is not a nuisance to be worked around; it is the content. Geometry and algebra are related contravariantly, and every functor built later in the collection inherits that orientation, which is why the D-module operation attached to a map $F$ that goes forward is called the inverse image.

Why the Jacobian is the natural first invariant

The Jacobian matrix is the best linear approximation to $F$ at each point, assembled into a single matrix with polynomial entries. Over $ℝ$ or $ℂ$ the inverse function theorem says that non-vanishing of $Δ F$ at $p$ gives an analytic local inverse near $p$ ; the algebraic shadow of that statement is the local inversion theorem for formal power series, which needs only $Δ F(0) ≠ 0$ . The difficulty, and the whole of the Jacobian conjecture, is that local inverses at every point do not obviously assemble into one global inverse, and even if they do the result need not be polynomial.

Construction and Proof

From homomorphisms back to maps

Suppose a $K$ -algebra homomorphism $φ : K[Y] → K[X]$ is given. Each $φ(y_i)$ is a polynomial in $K[X]$ , so the $m$ -tuple $(φ(y_1), … ,φ(y_m) )$ is the list of coordinate functions of a polynomial map, which we call $φ_* : X → Y$ . The two constructions are mutually inverse.

The dictionaryCoutinho (4.1.1), (4.1.2)

Let $F : X \to Y$ and $G : Y \to Z$ be polynomial maps and let $φ : K[Y] → K[X]$ , $ψ : K[Z] → K[Y]$ be $K$ -algebra homomorphisms. Then

$(F^#)_* = F$ and $(φ_*)^# = φ$ ;
$G ∘ F$ is a polynomial map and $(G ∘ F)^# = F^# ∘ G^#$ ;
$(φ ∘ ψ)_* = ψ_* ∘ φ_*$ .

So $F ↦ F^#$ is a bijection between polynomial maps $X \to Y$ and $K$ -algebra homomorphisms $K[Y] → K[X]$ , and it reverses composition.

Proof

For the first half of (1), read $y_i$ as the function on $Y$ that returns the $i$ -th coordinate of a point. Then $F^#(y_i) = y_i ∘ F = F_i$ , so the coordinate functions of $(F^#)_*$ are $F_1, … ,F_m$ , that is, $(F^#)_* = F$ . For the second half, $φ_*$ has coordinate functions $φ(y_i)$ by construction, so $(φ_*)^#$ and $φ$ agree on the generators $y_1, … ,y_m$ of $K[Y]$ ; two $K$ -algebra homomorphisms agreeing on generators are equal.

For (2), $G ∘ F$ has coordinate functions $G_k(F_1, … ,F_m)$ , which are polynomials, so it is a polynomial map. For $g ∈ K[Z]$ , associativity of composition gives $(G ∘ F)^#(g) = g ∘ (G ∘ F) = (g ∘ G) ∘ F = F^# (G^#(g) )$ . Statement (3) follows from (1) and (2) by applying the bijection to both sides.

Isomorphism criterionCoutinho (4.1.3)

A polynomial map $F : X \to Y$ is a polynomial isomorphism if and only if $F^# : K[Y] → K[X]$ is an isomorphism of $K$ -algebras.

Why the Jacobian must be a non-zero constant

Necessity of the Jacobian condition

If $F : K^n → K^n$ is a polynomial isomorphism with inverse $G$ , then $Δ F$ is a non-zero element of $K$ .

Proof

Apply (4.4) to $G ∘ F = id$ , whose Jacobian matrix is the identity and whose Jacobian determinant is $1$ . This gives $(Δ G ∘ F) ⋅ Δ F = 1$ in $K[X]$ , so $Δ F$ is a unit. Since $K[X]$ is a polynomial ring over a field it is a domain in which degrees add, so a product of two polynomials is $1$ only if both have degree $0$ . Hence $Δ F ∈ K^ ×$ .

Rescaling one coordinate function by the constant $Δ F$ turns any such $F$ into a map with $Δ F = 1$ , which is why the conjecture is always stated with the normalisation $Δ F = 1$ and loses nothing by it.

Key Equations

The following identities are used constantly in this chapter and the next few.

F^#(g) = g ∘ F, F^#(y_i) = F_i.

(4.1)

The comorphism, and its values on the coordinate variables.

(G ∘ F)^# = F^# ∘ G^#.

(4.2)

Contravariant functoriality: composing maps corresponds to composing comorphisms in the reverse order.

J(G ∘ F)(p) = J(G) (F(p) ) ⋅ J(F)(p).

(4.3)

The chain rule, valid as an identity of matrices of polynomials.

Δ(G ∘ F) = ( Δ G ∘ F ) ⋅ Δ F.

(4.4)

Multiplicativity of the Jacobian determinant, obtained from (4.3) by taking determinants.

F a polynomial isomorphism ; ⟹ ; Δ F ∈ K^ × .

(4.5)

The necessary condition. Setting $G = F^-1$ in (4.4) makes $Δ F$ a unit of $K[X]$ , and the units of a polynomial ring over a field are the non-zero constants.

K[F_1, … , F_n] = im F^# ⊂ eq K[x_1, … , x_n].

(4.6)

The image of the comorphism is the subalgebra generated by the coordinate functions; surjectivity of $F^#$ means this containment is an equality.

Variable Definitions

$K$: The ground field, of characteristic zero unless stated otherwise; in particular it is infinite.
$X = K^n$ , $Y = K^m$: Affine spaces, the source and target of a polynomial map.
$K[X]$ , $K[Y]$: The polynomial rings $K[x_1, … ,x_n]$ and $K[y_1, … ,y_m]$ , the coordinate rings of $X$ and $Y$ .
$F_1, … , F_m$: The coordinate functions of $F$ , elements of $K[X]$ .
$F^#$: The comorphism of $F$ , a $K$ -algebra homomorphism from $K[Y]$ to $K[X]$ .
$φ_*$: The polynomial map from $X$ to $Y$ attached to a $K$ -algebra homomorphism $φ : K[Y] → K[X]$ , with coordinate functions $φ(y_1), … , φ(y_m)$ .
$J(F)$: The Jacobian matrix, of size $m \times n$ over $K[X]$ , with $(i,j)$ entry the partial derivative of $F_i$ with respect to $x_j$ .
$Δ F$: The Jacobian determinant $J(F)$ , defined when $m = n$ .
$K^ ×$: The non-zero elements of $K$ ; these are exactly the units of $K[X]$ .

Properties and Behaviour

Injectivity of the comorphism means algebraic independence. $F^#$ is injective precisely when $F_1, … ,F_m$ are algebraically independent over $K$ , since a non-zero $g$ in the kernel is exactly a non-trivial algebraic relation $g(F_1, … ,F_m) = 0$ .
Surjectivity of the comorphism means the coordinate functions generate. $F^#$ is surjective precisely when $K[F_1, … ,F_n] = K[x_1, … ,x_n]$ , by (4.6).
Polynomial isomorphisms of $K^n$ form a group. Written $Aut_K K[x_1, … ,x_n]$ on the algebra side, it contains the affine maps and the elementary (triangular) maps $x_i ↦ x_i + f(x_1, … , x_i, … ,x_n)$ ; the subgroup these generate is called the tame subgroup.
The Jacobian determinant is a crossed homomorphism. Identity (4.4) says $Δ$ is multiplicative up to composing with the inner map, so on the group of polynomial automorphisms $Δ$ takes values in $K^ ×$ and is a genuine group homomorphism there.
Degree is not preserved. For $n = 1$ an invertible polynomial map must be affine, because the derivative is a non-zero constant. For $n ≥ 2$ the coordinate functions of an automorphism can have arbitrarily large degree, as the triangular examples show.
Over $ℂ$ , injective implies bijective with polynomial inverse. This is a theorem of Bass, Connell and Wright (1982, Thm. 2.1), resting on the Ax-Grothendieck theorem; it is what makes injectivity a legitimate target when attacking the conjecture.

Examples and Special Cases

Linear maps

If $F(p) = Ap$ for a matrix $A$ over $K$ , then $J(F) = A$ and $Δ F = A$ , a constant. The comorphism sends $y_i$ to the $i$ -th row of $A$ applied to $x$ , and $F$ is a polynomial isomorphism exactly when $A ≠ 0$ . The Jacobian criterion is therefore exactly right for linear maps; the conjecture asks whether that persists.

Elementary (triangular) automorphisms

Let $f ∈ K[x_2, … ,x_n]$ and put $F(x_1, … ,x_n) = (x_1 + f(x_2, … ,x_n), x_2, … , x_n )$ . The Jacobian matrix is unipotent lower- or upper-triangular in the appropriate ordering, so $Δ F = 1$ , and the inverse is $(u_1 - f(u_2, … ,u_n), u_2, … , u_n )$ . These maps show that automorphisms of $K^n$ , $n ≥ 2$ , can have coordinate functions of any degree.

The standard inclusion

For $n < m$ let $F(x_1, … ,x_n) = (x_1, … ,x_n,0, … ,0)$ . Then $F^#$ sends $g(y_1, … ,y_m)$ to $g(x_1, … ,x_n,0, … ,0)$ : it is surjective, with kernel the ideal generated by $y_n+1, … ,y_m$ . This is the algebraic form of "restrict a function to a coordinate subspace", and it reappears as the standard embedding when D-modules are transported along maps.

Non-constant Jacobian: the map $x ↦ x^2$

For $F : K \to K$ , $F(x) = x^2$ , we get $Δ F = 2x$ , which is not a unit. So $F$ is not a polynomial isomorphism, and indeed $K[x^2] ⊂ neq K[x]$ : the map is two-to-one over $ℂ$ away from the origin. Note that $F^#$ is still injective, so injectivity of the comorphism is far weaker than invertibility.

Bijective but not polynomially invertibleCoutinho, Ch. 4 Ex. 5.1

Over $K = R$ , let $F(x,y) = (x^3 + x,, y)$ . Then $Δ F = 3x^2 + 1 ≥ 1$ everywhere, so the Jacobian never vanishes on $R^2$ . The map $x ↦ x^3 + x$ is strictly increasing and surjective on $ℝ$ , so $F$ is a bijection of $R^2$ . Its inverse is not polynomial: if it were, $Δ F$ would have to be a non-zero constant by (4.5), and it is not. The inverse is given by Cardano's formula and is algebraic but not polynomial.

This is the example to keep in mind whenever the phrase "non-vanishing Jacobian" is used. Over a non-algebraically-closed field, non-vanishing at every rational point is strictly weaker than being a constant.

Worked Example

A non-triangular automorphism of $K^2$ and its comorphism

Step 1 - the map
Take $F : K^2 → K^2$ with coordinate functions

$F_1 = x + y^2, F_2 = y + (x + y^2)^2 = y + F_1^2.$

Neither coordinate function is left alone, so this is not a single elementary map; it is a composite of two of them, which is exactly how non-trivial automorphisms are usually manufactured.
Step 2 - compute the Jacobian determinant
Write $u = x + y^2$ for brevity. Then $∂ F_1/∂ x = 1$ , $∂ F_1/∂ y = 2y$ , $∂ F_2/∂ x = 2u$ and $∂ F_2/∂ y = 1 + 2u ⋅ 2y = 1 + 4uy$ . So

$J(F) = pmatrix 1 & 2y 2u & 1 + 4uy pmatrix, Δ F = (1 + 4uy) - 2y ⋅ 2u = 1.$

The cross terms cancel exactly, so $Δ F = 1$ identically, not merely at most points.
Step 3 - invert it
Set $u = F_1$ , $v = F_2$ . From $v = y + u^2$ we get $y = v - u^2$ ; substituting into $u = x + y^2$ gives $x = u - (v - u^2)^2$ . So the candidate inverse is

$G(u,v) = ( u - (v - u^2)^2, ; v - u^2 ),$

which is a polynomial map. Its Jacobian is $∂ G_1/∂ u = 1 + 4u(v-u^2)$ , $∂ G_1/∂ v = -2(v-u^2)$ , $∂ G_2/∂ u = -2u$ , $∂ G_2/∂ v = 1$ , and $Δ G = (1 + 4u(v-u^2) ) - 4u(v-u^2) = 1$ , consistent with (4.4).
Step 4 - check at a point
Take $(x,y) = (1,2)$ . Then $u = 1 + 4 = 5$ and $F(1,2) = (5,, 2 + 25) = (5, 27)$ . Running $G$ on $(5,27)$ : $v - u^2 = 27 - 25 = 2$ , and $u - 2^2 = 5 - 4 = 1$ , so $G(5,27) = (1,2)$ . The point comes back.
Step 5 - read the algebra
On the comorphism side, $F^#(y_1) = x + y^2$ and $F^#(y_2) = y + (x+y^2)^2$ . Surjectivity is visible directly: the subalgebra $K[F_1, F_2]$ contains $F_2 - F_1^2 = y$ , hence also $F_1 - y^2 = x$ , hence all of $K[x,y]$ . By the isomorphism criterion (4.1.3) this alone proves $F$ is a polynomial isomorphism, without ever writing down $G$ .

Result

$F(x,y) = (x + y^2,; y + (x+y^2)^2 )$ has $Δ F = 1$ and is a polynomial automorphism of $K^2$ , with inverse $G(u,v) = (u - (v-u^2)^2,; v - u^2 )$ . Equivalently $K[F_1,F_2] = K[x,y]$ , which is the reformulation of invertibility that the Jacobian conjecture attacks.

Applications and Industry Use

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

Transporting D-modules. The comorphism is the ring map along which modules are pulled back; see the ring map induced by a polynomial map and inverse images. Without the dictionary of this page there is no functorial D-module theory.
Automorphisms and twisted modules. A polynomial automorphism of $K^n$ lifts to an automorphism of the Weyl algebra, and twisting a module by an automorphism is one of the cheapest ways to manufacture new modules from old.
Change of variables in differential equations. Rewriting a system in new polynomial coordinates is exactly precomposition with a polynomial automorphism; the Jacobian determinant is the factor that appears when densities or integrals are transported.
Dynamical systems. Global questions about polynomial vector fields, treated in polynomial vector fields and the global asymptotic stability problem, are stated with the Jacobian matrix of the field, and the D-module attack on them in Chapter 19 begins with the material of this page.
Cryptography and coding. Polynomial maps with prescribed invertibility, and the difficulty of inverting them, underlie multivariate public-key schemes; there the ground field is finite, so the caveats in the failure-modes section are the whole story rather than a footnote.

Design Considerations

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

Which field. Take $K$ algebraically closed of characteristic zero if you want geometry and algebra to agree without qualification: points of $K^n$ then correspond to maximal ideals of $K[X]$ , and non-vanishing of $Δ F$ is the same as $Δ F ∈ K^ ×$ . Take $K = R$ only if the real-point structure matters, and expect the two hypotheses to separate.
Which direction to compute in. Deciding invertibility of $F$ directly means solving for an inverse map; deciding surjectivity of $F^#$ means a subalgebra membership test. The second is a finite, decidable computation for each fixed target, which is why (4.6) is the practical formulation.
Normalisation. Always rescale so that $Δ F = 1$ . Nothing is lost, and it removes a spurious constant from every subsequent formula.
Coordinates versus intrinsic language. The Jacobian matrix is coordinate-dependent; its determinant is not, up to the units that (4.4) tracks. If a construction survives composition with automorphisms, state it with $Δ F$ , not with $J(F)$ .

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: $ℕ$ , $ℝ$ , $ℂ$ for the number systems, $∂$ for partial differentiation, upright roman type for operator names such as $det$ and $deg$ , and the convention that the Jacobian matrix of a map has one row per component function.
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.
Citation numbering follows the source: results are referred to as chapter-section-item, so the dictionary theorem is $(4.1.1)$ and the isomorphism criterion is $(4.1.3)$ .
Terminology is not standardised across the literature. Coutinho writes the comorphism with a sharp superscript, $F^#$ , for what is elsewhere called the pullback $F^*$ or the comorphism; algebraic geometry texts often write $φ^#$ for the sheaf-level map. All denote the same construction here.

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 $Δ F$ is a symbolic determinant of an $n \times n$ matrix of polynomials. Expansion by minors costs $O(n!)$ and is unusable past small $n$ ; fraction-free Gaussian elimination (Bareiss) over the polynomial ring is the standard alternative, and interpolation on random points is often faster still, since $Δ F ≤ ∑ _i ( F_i - 1)$ bounds the degree to be interpolated.
Deciding $K[F_1, … ,F_n] = K[x_1, … ,x_n]$ is a subalgebra membership problem. The standard method introduces new variables $u_i$ , computes a Gröbner basis of the ideal generated by $u_i - F_i$ under an elimination order, and reduces each $x_j$ ; SAGBI bases give an alternative when the subalgebra has a finite canonical generating set.
Inverting a map when the inverse exists can be done by the same elimination: a reduced Gröbner basis of the ideal generated by $u_1 - F_1, … , u_n - F_n$ in $K[x,u]$ under an order eliminating the $x$ variables contains, when $F$ is invertible, relations $x_j - G_j(u)$ that exhibit the inverse. The degree of the inverse is bounded by $( F)^,n-1$ (Bass-Connell-Wright), which is sharp enough to be useful and large enough to make naive search hopeless.
Software. Singular, Macaulay2 and SageMath all provide polynomial ring arithmetic, Jacobian matrices and elimination orders; Singular ships a dedicated library for polynomial automorphisms and Jacobian problems. See Singular and Macaulay2.
Practical warning. Elimination Gröbner bases for maps of degree $3$ in $4$ or more variables routinely exhaust memory. Verifying a claimed inverse by composition is cheap; finding one is not.

Limits of Validity

The ground field must be infinite for a polynomial to be recoverable from the function it induces. Characteristic zero is the hypothesis used in the source, but infiniteness is the property actually needed.
Homomorphisms must be $K$ -algebra homomorphisms. The bijection of (4.1.1)-(4.1.2) fails for arbitrary ring homomorphisms as soon as $K$ has non-trivial field automorphisms.
$Δ F$ needs $m = n$ . For $m ≠ n$ there is no determinant; the substitute is the rank of $J(F)$ at a point, which controls smoothness and dominance but not invertibility.
Necessity is not sufficiency. Condition (4.5) is a genuine obstruction, so it can only ever disprove invertibility. Establishing invertibility from it is the open problem.
Everything here is about affine space. The same dictionary works for affine varieties, with reduced finitely generated $K$ -algebras in place of polynomial rings and closed embeddings in place of surjections of rings; see differential operators on an affine variety. Non-reduced or non-affine settings need scheme language, which this collection does not use.

Failure Modes and Common Mistakes

"Characteristic zero" is stronger than what is needed

The statement that a polynomial inducing the zero function is zero is usually justified by " $K$ has characteristic zero". The property genuinely used is that $K$ is infinite. Over $F_q$ the non-zero polynomial $x^q - x$ vanishes at every point of $F_q$ , so the map $K[Y] → functions$ has a kernel and comorphisms are no longer determined by the underlying set map. Every result on this page should be read as requiring $K$ infinite; characteristic zero is then imposed separately, for the differential calculus.

Ring homomorphism versus $K$ -algebra homomorphism

The correspondence $φ ↦ φ_*$ is stated for homomorphisms of polynomial rings, and it is easy to read that as "ring homomorphism". It is not enough. Let $K = C$ , $n = m = 1$ , and let $φ : C[y] → C[x]$ conjugate the coefficients and send $y \mapsto x$ . This is a ring isomorphism, but $φ(y) = x$ gives $φ_* = id$ and hence $(φ_*)^# = id ≠ φ$ . The bijection holds only for homomorphisms that fix $K$ pointwise.

Bijective is not the same as invertible in the polynomial category

A polynomial bijection of $K^n$ need not have a polynomial inverse; $F(x,y) = (x^3+x, y)$ over $ℝ$ is the standard witness. Over $ℂ$ the two notions do coincide, but that is a theorem (Bass-Connell-Wright, Thm. 2.1), not a definition, and it uses the algebraic closedness of $ℂ$ .

Non-vanishing at every point is weaker than being constant

Over $ℂ$ , a polynomial with no zeros in $C^n$ is a non-zero constant by the Nullstellensatz, so " $Δ F$ never vanishes" and " $Δ F ∈ K^ ×$ " agree. Over $ℝ$ they do not: $3x^2+1$ has no real zero and is not constant. Statements about the real case must say which of the two hypotheses is meant, and the honest real analogue of the conjecture is false (see the conjecture page for Pinchuk's counterexample).

Transposing the Jacobian breaks the chain rule

The convention here is rows indexed by coordinate functions, columns by source variables, so that (4.3) reads as matrix multiplication in the given order. Some sources use the transpose. The determinant is unaffected, so nothing in the conjecture changes, but any computation that keeps the matrix rather than its determinant, such as the construction of the derivations $D_i$ on the Dixmier conjecture page, must fix a convention and hold to it.

Historical Notes

The correspondence between affine varieties and finitely generated reduced algebras is due to the Hilbert-Noether generation of commutative algebra in the early twentieth century, and became standard through the work of Zariski and Weil; the polynomial-ring case recorded here is its most elementary instance.

The Jacobian condition entered the subject with Ott-Heinrich Keller in 1939, who asked whether an integral polynomial self-map of the plane with Jacobian determinant $1$ has an integral inverse. The conjecture as now stated grew out of that question.

Interest in the group of polynomial automorphisms itself is older still in dimension two, where Jung (1942) and van der Kulk (1953) proved that every automorphism of $K^2$ is tame, that is, a composite of affine and elementary maps. In dimension three the analogous statement is false: Nagata proposed a candidate wild automorphism in 1972, and Shestakov and Umirbaev proved in 2004 that it is indeed not tame. That failure is one reason the Jacobian conjecture resists structural attacks for $n ≥ 3$ .

The 1982 survey of Bass, Connell and Wright collected what was known, supplied the reduction to degree three and the degree bound on inverses, and is still the standard entry point to the literature.

Comparison

The geometry-algebra dictionary for affine spaces.
Geometric side	Algebraic side	Comment
Affine space $X = K^n$	Polynomial ring $K[x_1, … ,x_n]$	Points correspond to maximal ideals when $K$ is algebraically closed
Polynomial map $F : X \to Y$	$K$ -algebra map $F^# : K[Y] → K[X]$	Arrow reverses
Composition $G ∘ F$	$F^# ∘ G^#$	Contravariant functor
$F$ has algebraically independent components	$F^#$ injective	Equivalent to $F^# = 0$
Components generate all polynomials	$F^#$ surjective	This is identity (4.6)
$F$ is a polynomial isomorphism	$F^#$ is an algebra isomorphism	Corollary (4.1.3)
Closed embedding of a coordinate subspace	Surjection with kernel a monomial ideal	The standard embedding
Local invertibility near a point	$Δ F(p) ≠ 0$	Only over $ℝ$ , $ℂ$ , or formally; see the local inversion theorem

Key Takeaways

Key takeaways

A polynomial map $F : K^n → K^m$ is exactly the data of a $K$ -algebra homomorphism $F^# : K[y_1, … ,y_m] → K[x_1, … ,x_n]$ , with $F^#(y_i) = F_i$ ; the assignment reverses composition.
$F$ is a polynomial isomorphism if and only if $F^#$ is an algebra isomorphism, so invertibility becomes a statement about generation of a polynomial ring.
The chain rule makes $Δ$ multiplicative, so an invertible $F$ has $Δ F ∈ K^ ×$ ; after rescaling one may always assume $Δ F = 1$ .
The converse is the Jacobian conjecture and is open for every $n ≥ 2$ . Necessity is elementary; sufficiency is not.
The identification of polynomials with functions needs $K$ infinite, and the dictionary needs homomorphisms that fix $K$ ; both hypotheses are silently used and both can fail.
Non-vanishing of $Δ F$ at every point of $K^n$ is equivalent to $Δ F$ being constant only when $K$ is algebraically closed.

FAQs

Why is the comorphism written with the arrow reversed?

Because functions pull back. If $g$ is a function on the target $Y$ and $F$ maps $X$ into $Y$ , then $g ∘ F$ is a function on $X$ . There is no way to push a function forward along a non-injective map without extra choices, so the natural operation goes from $K[Y]$ to $K[X]$ .

Does

Δ F = 1

imply

F

is injective?

It implies that the comorphism $F^#$ is injective, equivalently that $F_1, … ,F_n$ are algebraically independent; that is proved on the conjecture page and needs only $Δ F ≠ 0$ everywhere. Injectivity of $F$ itself as a map of points is known over $ℂ$ only in special cases, for instance when all coordinate functions have degree at most $2$ .

If the Jacobian never vanishes, does the inverse function theorem not finish the job?

No. It gives a local inverse near each point, defined only on a neighbourhood and only analytic. Two obstacles remain: the local inverses must agree on overlaps and cover the whole space, which is a global topological question, and the resulting global inverse must be polynomial, which the theorem says nothing about. The example $x ↦ x^3 + x$ over $ℝ$ clears the first hurdle and fails the second.

Is the Jacobian conjecture just the statement that

Δ F

is a unit?

The other way round. That $Δ F$ is a unit is the easy, proved direction. The conjecture is the converse: that $Δ F ∈ K^ ×$ forces the existence of a polynomial inverse.

What changes in positive characteristic?

Two things break. Polynomials are no longer determined by their values on $K^n$ when $K$ is finite, and the differential calculus loses information because $p$ -th powers have zero derivative. The map $F(x_1, … ,x_n) = (x_1 + x_1^,p, x_2, … , x_n)$ over a field of characteristic $p$ has Jacobian matrix the identity yet is not invertible, since $K[x_1 + x_1^,p]$ contains no element of degree $1$ . See the positive characteristic page for the parallel failure on the operator side.

How do I actually check whether a given map is invertible?

Compute $Δ F$ first; if it is not a non-zero constant you are done, the map is not invertible. If it is, try to express each $x_j$ as a polynomial in $F_1, … ,F_n$ , either by hand as in the worked example or by an elimination Gröbner basis. Success proves invertibility; failure proves nothing, because the search may simply have been out of reach.

Why do polynomial maps matter for D-modules at all?

Because the functors that make the theory useful are attached to maps. The inverse image of a D-module along $F$ is built from the comorphism $F^#$ together with the chain rule, which is precisely the Jacobian matrix telling you how derivations transform. The material on this page is the input to that construction.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 4 §1, results (4.1.1), (4.1.2), (4.1.3), and Exercises 5.1-5.3.
O.-H. Keller, Ganze Cremona-Transformationen, Monatshefte für Mathematik und Physik 47 (1939), 299-306 - the origin of the Jacobian condition.
H. Bass, E. H. Connell and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bulletin of the American Mathematical Society 7 (1982), 287-330 - Theorem 2.1 on injective polynomial maps of complex affine space, the degree bound on inverses, and the reduction to degree three.
H. W. E. Jung, Über ganze birationale Transformationen der Ebene, Journal für die reine und angewandte Mathematik 184 (1942), 161-174, and W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde 1 (1953), 33-41 - tameness in dimension two.
I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, Journal of the American Mathematical Society 17 (2004), 197-227 - Nagata's automorphism is wild.
A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics 190, Birkhäuser, 2000 - the standard modern monograph, including the Gröbner-basis algorithms mentioned above.
ISO 80000-2, Quantities and units - Part 2: Mathematics - notation for number systems, partial derivatives and matrix operations.
ISO/IEC 40314, Information technology - Mathematical Markup Language (MathML) Version 3.0 - the encoding used for the mathematics on this page.
Singular team, Singular computer algebra system documentation - polynomial ring arithmetic, Jacobian matrices, elimination orders and libraries for polynomial automorphisms.

AI Suggested Questions

Prove in detail that a polynomial vanishing at every point of $K^n$ is zero when $K$ is infinite, and exhibit the failure over $F_2$ for $n = 2$ .
Show that the units of $K[x_1, … ,x_n]$ are exactly $K^ ×$ , and explain where the domain property is used.
Write down the composite of two elementary automorphisms of $K^3$ and compute its Jacobian determinant directly.
Give a polynomial map $K^2 → K^2$ with $Δ F$ a non-zero constant that you can prove is not injective, or explain why no such example is known.
Compute the comorphism of the graph embedding $x ↦ (x, f(x))$ and identify its kernel.
Explain how the degree bound $( F)^,n-1$ on the inverse is used to make invertibility a finite computation.
Verify that Nagata's map on $K^3$ has Jacobian determinant $1$ , and state precisely what tameness would have given.