On endomorphisms of affine spaces and the Jacobian problem

Abstract: Let $p$ be a prime. We provide examples which show that étale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\in\mathbb N\times\mathbb N<sup>{\ast}$. We provide examples of such étale endomorphisms whose images have complements of cardinality $l$ and whose geometric degrees are $pm$. Several conjectures are disproved, and in particular we provide an analog over $k$ of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each $k$ are included in dimension $2$ (resp.\ in all dimensions at least $3$); for instance, if $p=2$, then we show for each $m$ there exist surjective étale endomorphisms of the affine spaces over $k$ of dimension at least $3$ of geometric degree $m$. If $e:X\rightarrow X$ is an endomorphism of a variety over an algebraically closed field $K$, then we show that there exists $n\in\mathbb N$ such that $\Imm(e<sup>n)=\Imm(e<sup>{n+1})$ provided either (i) $e$ is quasi-finite or (ii) $\dim(X)\le 2$ and $X\setminus\Imm(e)$ is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over $K$ whose images have complements of cardinality $l$ and for all $n\in\mathbb N$ we have $\Imm(e<sup>n)\neq\Imm(e<sup>{n+1})$. We prove that all affine moduli schemes of étale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over $K$ are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over $\mathbb C$ and Adjamagbo's analog of it over $k$ hold for étale endomorphisms of affine spaces that are composites $g\circ f$, where $f$ is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and $g$ is a projection that omits one coordinate.

• The paper introduces a comprehensive framework of invariants, including geometric and algebraic degrees, fiber cardinalities and geometric invariants to analyze the structure of étale endomorphisms.
• Disproves that there are fundamental limitations in positive characteristic settings and that Había counterexamples contradict assumptions.
• Provides comprehensive modularity on maps.

Scope and mathematical setting

The paper studies polynomial self-maps of affine space whose Jacobian determinant is a nonzero constant, with particular emphasis on étale endomorphisms over algebraically closed fields of positive characteristic. Such maps are called Kraus–Keller maps. The central issue is that the Jacobian condition controls infinitesimal behavior but, unlike in characteristic zero under the conjectural Jacobian principle, does not force global invertibility or even surjectivity in positive characteristic.

The authors introduce a systematic collection of invariants for an étale endomorphism

$e:\mathbb A^n_K\longrightarrow \mathbb A^n_K.$

These include the geometric degree, algebraic degrees up to left, right, or two-sided composition by automorphisms, the possible cardinalities of non-generic fibers, the cardinality of the complement of the image, stabilization behavior under iteration, and geometric invariants obtained from the normalization of the target in the source function field. The latter construction associates to $e$ a normal variety $X_e$ containing the source affine space as an open subvariety and a finite self-map

$\psi_e:X_e\longrightarrow X_e,$

called the associated Jacobian variety. The irreducible components of $X_e\setminus\mathbb A^n$ define the class-rank invariants $\rho(e)$ and $\rho_{\mathrm{\acute et}}(e)$.

This framework permits the paper to treat global image defects, fiber degenerations, iteration, normalization, singularities, and moduli-theoretic properties within a single geometric formalism. The main results are constructive, although some constructions are presented existentially after passing through normalizations, finite covers, or affine models.

Frobenius systems and the failure of positive-characteristic surjectivity

A basic endomorphism in characteristic $p$ is, up to polynomial automorphisms, a map of the form

$$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$

Its Jacobian matrix is the identity, so every basic endomorphism is étale. The associated preimage problem for the origin is the Frobenius system

$x_i+f_i(x_1^p,\ldots,x_n^p)=0.$

A motivating conjecture proposed that Frobenius systems over algebraically closed fields should always be consistent. The paper disproves this in dimension two by constructing explicit inconsistent systems. In particular, the authors exhibit a basic étale endomorphism of $e$0 whose image omits a point. Thus even the positive-characteristic analogue of the elementary surjectivity statement for one-variable maps fails in dimension two.

The construction is not merely a pathological example. It produces Frobenius systems with controlled numbers of solutions. Theorem 1(1) shows that for every $e$1 there is an $e$2-system with exactly $e$3 solutions, including the case $e$4. Consequently, for every prime $e$5, there are étale endomorphisms of the affine plane having fibers of arbitrary prescribed finite cardinalities below their generic degree. This is a strong obstruction to any positive-characteristic surjectivity principle based only on the Jacobian condition.

The construction uses affine surfaces and open embeddings rather than only direct manipulation of polynomial equations. A key geometric ingredient is the surface

$e$6

together with explicit open embeddings of $e$7 into this surface. The complement of the image can be arranged to be a prescribed disjoint union of affine lines and points. Finite étale projections from these surfaces then yield endomorphisms of $e$8 with controlled degree, image complement, and normalization geometry.

Arbitrary gaps, degrees, and fiber structures in dimension two

Theorem 1(2) gives the most precise numerical family in the paper. If $e$9 and $X_e$0 satisfy $X_e$1, there exists an étale endomorphism $X_e$2 of $X_e$3 defined over $X_e$4 with

$X_e$5

where $X_e$6 is the cardinality of the complement of the image. The associated Jacobian variety is étale, and

$X_e$7

The left and right algebraic degrees satisfy

$X_e$8

with $X_e$9 bounded above by the same quantity.

The identity

$\psi_e:X_e\longrightarrow X_e,$0

follows. Thus, even while fixing the geometric degree to a multiple of $\psi_e:X_e\longrightarrow X_e,$1, one can prescribe an arbitrary finite image defect. The implication is that geometric degree and surjectivity are largely independent invariants in positive characteristic.

Theorem 1(3) supplies a complementary family. If $\psi_e:X_e\longrightarrow X_e,$2 divides $\psi_e:X_e\longrightarrow X_e,$3, there are étale endomorphisms of degree $\psi_e:X_e\longrightarrow X_e,$4 with

$\psi_e:X_e\longrightarrow X_e,$5

When $\psi_e:X_e\longrightarrow X_e,$6, the iteration index is $\psi_e:X_e\longrightarrow X_e,$7; when $\psi_e:X_e\longrightarrow X_e,$8, necessarily $\psi_e:X_e\longrightarrow X_e,$9, and $X_e\setminus\mathbb A^n$0. The associated Jacobian variety is regular, but it is non-étale for $X_e\setminus\mathbb A^n$1. This separates regularity of the normalization from étaleness of the associated self-map.

The paper also obtains explicit examples with arbitrary finite fiber cardinalities. In particular, the non-generic fiber set $X_e\setminus\mathbb A^n$2 can contain several distinct values, including zero-dimensional fibers, one-point fibers, and fibers of cardinality close to the generic degree. The examples show that fiber stratification in the positive-characteristic étale category is substantially more flexible than suggested by characteristic-zero intuition.

Counterexamples to positive-characteristic Jacobian conjectures

The paper disproves Adjamagbo’s characteristic-$X_e\setminus\mathbb A^n$3 Jacobian Conjecture, which asserted that an étale endomorphism of affine $X_e\setminus\mathbb A^n$4-space with geometric degree not divisible by $X_e\setminus\mathbb A^n$5 should be an automorphism. The conjecture fails in every dimension $X_e\setminus\mathbb A^n$6.

In dimension two, the authors construct families with degrees not divisible by $X_e\setminus\mathbb A^n$7, using substitution–étale tame perturbations. In dimension at least three, the linear–tame–perturbation method produces broader families, including both surjective and non-surjective examples. These families are obtained by starting with an étale map in two variables, adjoining a trivial coordinate, composing with a tame automorphism, and perturbing one coordinate by a $X_e\setminus\mathbb A^n$8-th power. Because the perturbation does not change the Jacobian matrix, étaleness is preserved, while the induced function-field extension can acquire degrees outside $X_e\setminus\mathbb A^n$9.

The resulting contradiction to Adjamagbo’s conjecture is particularly explicit in characteristic two. For every $\rho(e)$0 and every dimension $\rho(e)$1, the paper constructs a surjective étale endomorphism of geometric degree $\rho(e)$2. It also constructs non-surjective examples for broad classes of degrees. Therefore, in characteristic two, surjectivity does not constrain the geometric degree in the way proposed by the conjecture.

The dimension-two situation remains more constrained. The paper proves that for $\rho(e)$3, every positive integer occurs as the geometric degree of an étale endomorphism of $\rho(e)$4. For general $\rho(e)$5, however, the authors do not classify all possible degrees prime to $\rho(e)$6. They formulate this classification explicitly as an open problem. The weak Jacobian conjecture proposed in the paper remains unresolved: for fixed $\rho(e)$7, it asks whether sufficiently large characteristic excludes degrees in the interval from $\rho(e)$8 through $\rho(e)$9 for étale endomorphisms of the affine plane.

Iteration and stabilization of images

A second theme concerns the sequence

$\rho_{\mathrm{\acute et}}(e)$0

for an endomorphism of a variety. The stabilization index $\rho_{\mathrm{\acute et}}(e)$1 is the least $\rho_{\mathrm{\acute et}}(e)$2 such that $\rho_{\mathrm{\acute et}}(e)$3, if such an index exists.

The paper proves that if $\rho_{\mathrm{\acute et}}(e)$4 is quasi-finite, then $\rho_{\mathrm{\acute et}}(e)$5 is finite. This result applies over arbitrary algebraically closed fields and is obtained by reduction to positive characteristic. It generalizes earlier results that required stronger hypotheses, such as affine varieties over $\rho_{\mathrm{\acute et}}(e)$6 and finite image complements.

For an arbitrary variety $\rho_{\mathrm{\acute et}}(e)$7 of dimension at most two, if $\rho_{\mathrm{\acute et}}(e)$8 is finite, then image stabilization also occurs. The proof uses the monotonicity of the successive increments

$\rho_{\mathrm{\acute et}}(e)$9

which form a non-increasing sequence. In particular, if the image complement of $p$0 is finite, then the complement of every iterate remains finite and grows in a controlled manner.

The behavior changes sharply in dimension at least three. The authors construct endomorphisms whose image complements have prescribed finite cardinality but for which

$p$1

for every $p$2. Thus quasi-finiteness or finite image complement is essential in the stabilization theorems. The examples show that finite initial defect does not by itself imply eventual stabilization once the dimension is sufficiently large.

The set-theoretic estimates are also structurally informative. If the smallest nonempty fiber has cardinality $p$3, then the image-defect sequence is bounded by a geometric-series estimate, yielding an explicit bound on $p$4. Infinite iteration index forces $p$5 for every positive iterate. Hence persistent non-stabilization can occur only through the repeated appearance of one-point fibers.

Geometry of the associated Jacobian varieties

For an étale endomorphism $p$6, the normalization $p$7 of the target in the source function field compactifies the source relative to the finite morphism induced by $p$8. The open immersion

$p$9

has complement of pure codimension one, and the induced finite self-map $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$0 encodes the failure of $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$1 to be finite.

In dimension two, every irreducible component of the complement is shown to be isomorphic to $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$2. The associated affine surface is therefore a sphere-like surface: a normal affine surface containing an open copy of $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$3 whose complement is irreducible. Such surfaces admit an $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$4-fibration over $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$5, with the embedded affine plane equal to the inverse image of $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$6.

Theorem 10 establishes several strong properties of sphere-like surfaces. The boundary curve is rational and has one point at infinity; in fact, it is isomorphic to $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$7. There is at most one singular point. Every singularity is rational, and in characteristic zero a singularity is log-terminal. If the unique non-reduced fiber has multiplicity prime to the characteristic, the singularity is a quotient singularity by a subgroup of a cyclic group. In the wild case, it is still a quotient singularity by a finite group, although the group need not be cyclic.

These results provide geometric control over the normalization spaces attached to étale endomorphisms of the affine plane. The canonical affine cover by sphere-like surfaces decomposes according to the behavior of $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$8: components in the étale locus give regular sphere-like surfaces, while components outside it give quasi-spheres or pseudo-spheres. The numbers of these components are measured by the refined class-rank invariants.

The tangent bundle criterion is especially significant. For an affine open subvariety $$(x_1,\ldots,x_n)\longmapsto \bigl(x_1+f_1(x_1^p,\ldots,x_n^p),\ldots, x_n+f_n(x_1^p,\ldots,x_n^p)\bigr).$$9 containing the embedded affine space, the following are equivalent: $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$0 lies in the étale locus of $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$1; $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$2 is regular with trivial tangent bundle; and $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$3 is a complete intersection. In characteristic $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$4, these are also equivalent to the coordinate ring possessing a $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$5-basis or a differential basis. If $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$6 is regular but non-étale, then $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$7 cannot be a complete intersection. This gives a geometric obstruction to complete-intersection presentations that is detected by the ramification of the associated Jacobian variety.

Finite étale covers and torsorial constructions

The paper constructs finite étale covers of $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$8 in dimensions $x_i+f_i(x_1^p,\ldots,x_n^p)=0.$9 and $e$00 using pullbacks of Lang torsors associated with $e$01. The resulting covers are defined over $e$02 and factor through finite étale morphisms of degrees

$e$03

For $e$04 and $e$05, the construction gives an intermediate rational cover of degree $e$06 with a further Galois cover of degree $e$07. For $e$08 and $e$09, there is an intermediate rational cover of degree $e$10 with a further quaternionic Galois cover of order $e$11. Since the intermediate varieties have only constant units, these covers imply non-factoriality through Kummer theory.

The dimension-two examples arise as quotients of the dimension-three constructions by a left $e$12-action. This provides explicit positive-characteristic counterexamples to conjectures asserting that certain rational affine varieties or affine spaces have trivial prime-to-$e$13 étale fundamental group. It also demonstrates that finite étale covers of affine space in characteristic $e$14 can arise from highly nontrivial torsorial geometry while retaining rationality of the covering varieties.

Moduli of Jacobian maps

The authors study affine moduli schemes parametrizing étale endomorphisms of $e$15 defined by polynomial tuples of bounded degree and with Jacobian determinant equal to $e$16. For every degree bound $e$17, the corresponding moduli affine scheme is connected.

The smoothness classification is unusually rigid. The moduli scheme is smooth precisely when

$e$18

or in the exceptional case

$e$19

Thus bounded-degree spaces of Jacobian maps are generally connected but singular. The implication is that connectedness does not reflect a simple geometric interpolation between automorphisms and non-invertible maps: the parameter spaces retain singularities except in low-degree or characteristic-two exceptional cases.

The paper also disproves additional conjectures concerning the geometry of these moduli spaces. Because the parameter spaces are affine schemes rather than merely sets of polynomial tuples, the result provides deformation-theoretic information about families of Jacobian maps, although it does not yield a classification of their irreducible components beyond connectedness.

Structural factorization results

One positive result concerns a restricted factorization class. Suppose an étale endomorphism factors as

$e$20

where the first map is quasi-finite and a locally closed embedding in codimension one away from a finite set, and the second map is a coordinate projection. The paper proves that the classical Jacobian Conjecture holds for such maps in characteristic zero. The corresponding characteristic-$e$21 degree restriction also holds in positive characteristic.

This result isolates a geometric mechanism under which the conjectural conclusion survives: the first factor has sufficiently controlled codimension-one behavior, while the second factor is a projection. It does not address arbitrary étale endomorphisms, and the finite exceptional subset in the embedding hypothesis is essential to the stated argument.

Limitations and open questions

The paper leaves several central classification problems unresolved. Most importantly, it does not settle the weak positive-characteristic Jacobian conjecture in dimension two. The constructions produce many degrees and arbitrary finite image gaps, but they do not determine the complete set of degrees prime to $e$22 that can occur for étale endomorphisms of $e$23.

The paper also does not resolve the surjectivity conjecture over $e$24 in dimension two. Its positive-characteristic examples cannot be transferred directly to characteristic zero, and the authors explicitly identify this as an open problem. Similarly, the classification of all possible sequences of image-complement cardinalities under iteration remains open, despite the general monotonicity and boundedness results.

The geometric classification of the normalizations $e$25 is also incomplete. When the associated Jacobian variety is regular but non-étale, the paper proves that $e$26 is not a complete intersection, but no general classification of these varieties is available. The hypersurface and sphere-like constructions describe substantial families rather than all possibilities.

Finally, the moduli results establish connectedness and characterize smoothness, but they do not describe the singularities, normalization, irreducible components, or deformation theory of the moduli schemes in general. These questions are especially relevant because singularity of the moduli space is the generic behavior outside the stated exceptional cases.

Conclusion

The paper develops a broad geometric and algebraic theory of étale endomorphisms of affine spaces, with emphasis on invariants that detect image defects, fiber cardinalities, iteration, normalization geometry, and singularities. Its principal conclusions are negative for several positive-characteristic Jacobian conjectures: étale maps can have arbitrary finite image gaps in dimension two, degrees not divisible by the characteristic, and highly nontrivial associated covers. At the same time, the paper proves positive structural results for quasi-finite iteration, sphere-like normalizations, complete-intersection criteria, moduli connectedness, and restricted factorizations. The unresolved dimension-two degree and surjectivity problems delimit the scope of the constructions and identify the main questions left by the theory (2609.05746).