Inverse degree bounds for polynomial automorphisms with bounded-rank Jacobian perturbation (Question 5.9 of Yan–de Bondt)

For a polynomial automorphism $F = X + H$ over a characteristic-zero field, where $X$ is the identity map and $H$ is the nonlinear part, Yan and de Bondt (J. Algebra 384, 2013, Question 5.9) ask whether $\deg(F^{-1}) \leq (\deg F)^k$ whenever the Jacobian matrix $JH$ of $H$ has rank at most $k$ over the rational function field. This would sharpen the classical Bass–Connell–Wright bound $\deg(F^{-1}) \leq (\deg F)^{n-1}$ for all polynomial automorphisms. The authors settle the extreme regimes (rank $JH \leq 1$ and rank $JH \geq n-1$, plus all maps satisfying a kernel-equality condition, including power-linear maps), but the middle regime $2 \leq k \leq n-2$ remains open. The sibling nilpotency-index version (Question 5.5 of the same paper) is true for $k \leq 2$ and refuted for $k \geq 3$ by a Furter example within the paper itself, so the rank version is the precise surviving open core. A positive answer would give new structural control over inverse degrees for the low-rank classes studied in the Jacobian-conjecture literature; a counterexample would show the rank of $JH$ does not control inverse degree.

Background

The Jacobian conjecture (JC), posed by Keller in 1939, asks whether every polynomial map $F: K^n \to K^n$ over a field $K$ of characteristic zero whose Jacobian matrix $JF$ (the matrix of partial derivatives $\partial F_i/\partial x_j$) has a nonzero constant determinant — a so-called Keller map — is invertible with a polynomial inverse. The conjecture was refuted in 2026 by an explicit counterexample in dimension $3$ with nonzero constant Jacobian determinant, which extends to every dimension $n \geq 3$; it now remains open only for $n = 2$. A web of reductions narrowed the class of maps that had to be checked. Bass, Connell and Wright (1982) reduced the conjecture to maps of the form $X + H$ with $H$ cubic and homogeneous, and proved in the same paper the general inverse-degree bound $\deg(F^{-1}) \leq (\deg F)^{n-1}$ valid for every polynomial automorphism, where $\deg F$ is the maximum total degree of the components of $F$. Drużkowski (1983) reduced the conjecture further to power-linear maps $X + (AX)^{*3}$, where $(AX)^{*3}$ is the componentwise cube of the linear map $AX$ with $A$ a square matrix; the rank of $A$ then measures the complexity of the map, and Drużkowski (1993) settled the cubic homogeneous case with rank or corank below three. De Bondt and van den Essen (2005) reduced the conjecture to the case of symmetric matrices $A$. Yan and de Bondt (2013) studied non-homogeneous power-linear maps, where $H_i = (A_i X)^{d_i}$ with possibly different exponents $d_i$ and $A$ the matrix with rows $A_i$; they proved (their Problems 4.3/4.5) that invertibility of all such Keller maps with $\mathrm{rank}(A) \leq r$ is equivalent to the JC in dimension $r$; since that conjecture now fails for every $r \geq 3$, the sibling statement is refuted for $r \geq 3$ and remains open only for $r = 2$. A complementary, independent line in the same paper concerns degrees of inverses. Write $JH$ for the Jacobian matrix of $H$ and $\mathrm{rank}\, JH$ for its rank over the rational function field $K(x_1,\dots,x_n)$. Two hypotheses on $H$ are natural: nilpotency of a given index, i.e. $JH^{k+1} = 0$, and small rank, i.e. $\mathrm{rank}\, JH \leq k$; for a nilpotent $JH$, rank at most $k$ forces $JH^{k+1} = 0$, so the rank hypothesis is the stronger one. Under the nilpotency hypothesis the authors prove $\deg(F^{-1}) \leq (\deg F)^k$ for homogeneous power-linear $F$ when $k \leq 2$ (their Question 5.5), but a counterexample of Furter (their Example 5.7, a quadratic map with $JH^4 = 0$ and $\deg(F^{-1}) \geq 12 > 8 = 2^3$) refutes that statement for $k \geq 3$. Replacing nilpotency by rank, they pose Question 5.9 — the subject of this record — and answer it affirmatively only in the extreme regimes: rank $JH \leq 1$ (via a normal form for $H$), rank $JH \geq n-1$ (where the Bass–Connell–Wright bound already gives the claim), and whenever $\ker JH = \ker_K JH$, where $\ker JH$ is taken over the rational function field and $\ker_K JH := (\ker JH) \cap K^n$ is the space of constant-vector solutions — a condition automatically satisfied by (non-homogeneous) power-linear maps. The middle-rank cases are left open.

Problem Statement

Let $K$ be a field of characteristic zero and let $F = X + H: K^n \to K^n$ be an invertible polynomial map (a polynomial automorphism), where $X = (x_1,\dots,x_n)$ is the identity map and $H = F - X$. Suppose that the Jacobian matrix $JH$ of $H$, regarded as a matrix over the rational function field $K(x_1,\dots,x_n)$, has rank at most $k$. Must $F$ satisfy the degree bound $\deg(F^{-1}) \leq (\deg F)^k$, where degrees are maximum total degrees of the components? This is Question 5.9 of Yan–de Bondt (2013). The answer is known to be affirmative when $\mathrm{rank}\, JH \leq 1$, when $\mathrm{rank}\, JH \geq n-1$, and whenever $\ker JH = \ker_K JH$, that is, $\dim(\ker JH) = \dim_K(\ker_K JH)$ where $\ker_K JH := (\ker JH) \cap K^n$ is the space of constant-vector solutions of $JH v = 0$ (for instance for all power-linear maps). The open core is the middle regime $2 \leq k \leq n-2$ (so $n \geq 4$): either prove the bound for every polynomial automorphism $F = X + H$ with $\mathrm{rank}\, JH \leq k$, or exhibit one such invertible map with $\deg(F^{-1}) > (\deg F)^k$.

References

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