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formal-inverse

(page created)

Revision 252 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__formal-inverse.md).”

Status and infobox

status: — → proved
tier: — → V3
kind: — → theorem
year: — → 1982
statement: — → In characteristic $0$, $F=X-H$ with $H$ of order at least 2 has a unique formal inverse $G=X+\sum_T\frac1{|\mathrm{Aut}\,T|}P_{T,H}$, summed over isomorphism classes of rooted trees. A Keller map is an automorphism if and only if $G$ is a polynomial, and by Gabber's bound $\deg F^{-1}\le(\deg F)^{n-1}$ that is a finite check.
short: — → The inverse of x − H is a sum over rooted trees; invertible ⇔ the sums stop

Relations

+ related to jc-n: For homogeneous $H$ of degree $d$ in $n$ variables, invertibility is the vanishing of the tree sums of every size m with (d−1)m + 1 > d^{n−1}, and through the cubic reduction the all-dimensions conjecture is this statement for cubic H. The approach was pursued by Wright, by Singer, and by Bisi and coauthors, through random planar trees and $d$ -Catalan trees.

Source +17 −0

+---

+title: The formal inverse and its tree formula

+short: The inverse of x − H is a sum over rooted trees; invertible ⇔ the sums stop

+kind: theorem

+status: proved

+tier: V3

+year: 1982

+authors: H. Bass, E. H. Connell and D. Wright (1982); D. Wright (1987, 1989); the Abhyankar–Gurjar inversion formula

+statement: In characteristic $0$, $F=X-H$ with $H$ of order at least 2 has a unique formal inverse $G=X+\sum_T\frac1{|\mathrm{Aut}\,T|}P_{T,H}$, summed over isomorphism classes of rooted trees. A Keller map is an automorphism if and only if $G$ is a polynomial, and by Gabber's bound $\deg F^{-1}\le(\deg F)^{n-1}$ that is a finite check.

+---

+This turns invertibility into combinatorics. For a Keller map, invertibility is the vanishing of the tree sums $\sum_{|T|=m}P_{T,H}/|\mathrm{Aut}\,T|$ for all large $m$. Bass, Connell and Wright proved the formula for homogeneous $H$ with $\det JF=1$, in Chapter III of their survey. Wright removed both hypotheses and derived two related tree formulas, one due to Towber. All of them come from the Abhyankar–Gurjar inversion formula.<ref>D. Wright, The tree formulas for reversion of power series, J. Pure Appl. Algebra 57 (1989) 191–211, [doi:10.1016/0022-4049(89)90116-3](https://doi.org/10.1016/0022-4049(89)90116-3); Formal inverse expansion and the Jacobian conjecture, J. Pure Appl. Algebra 48 (1987) 199–219.</ref>

+For homogeneous $H$ of degree $d$ in $n$ variables, [[related::Statement:jc-n|invertibility is the vanishing of the tree sums of every size m with (d−1)m + 1 > d^{n−1}, and through the cubic reduction the all-dimensions conjecture is this statement for cubic H]]. The approach was pursued by Wright, by Singer, and by Bisi and coauthors, through random planar trees and $d$-Catalan trees. Meng's tree formula for the Legendre transform yields the Abhyankar–Gurjar formula as a corollary.

+After 2026, the explicit [[Statement:cubic-homogeneous-counterexamples|cubic-homogeneous counterexamples]] have formal inverses that are not polynomials, so their tree sums are nonzero in infinitely many degrees. Zhao's proof of the [[Statement:zhao-vanishing|vanishing equivalence]] writes this formal inverse in terms of $\Delta^mP^{m+1}$.

+<references />

Revision 252 as rendered

This turns invertibility into combinatorics. For a Keller map, invertibility is the vanishing of the tree sums $\sum_{|T|=m}P_{T,H}/|\mathrm{Aut}\,T|$ for all large $m$ . Bass, Connell and Wright proved the formula for homogeneous $H$ with $\det JF=1$ , in Chapter III of their survey. Wright removed both hypotheses and derived two related tree formulas, one due to Towber. All of them come from the Abhyankar–Gurjar inversion formula.^[1]

For homogeneous $H$ of degree $d$ in $n$ variables, invertibility is the vanishing of the tree sums of every size m with (d−1)m + 1 > d^{n−1}, and through the cubic reduction the all-dimensions conjecture is this statement for cubic H. The approach was pursued by Wright, by Singer, and by Bisi and coauthors, through random planar trees and $d$ -Catalan trees. Meng's tree formula for the Legendre transform yields the Abhyankar–Gurjar formula as a corollary.

After 2026, the explicit cubic-homogeneous counterexamples have formal inverses that are not polynomials, so their tree sums are nonzero in infinitely many degrees. Zhao's proof of the vanishing equivalence writes this formal inverse in terms of $\Delta^mP^{m+1}$ .

↑ D. Wright, The tree formulas for reversion of power series, J. Pure Appl. Algebra 57 (1989) 191–211, doi:10.1016/0022-4049(89)90116-3; Formal inverse expansion and the Jacobian conjecture, J. Pure Appl. Algebra 48 (1987) 199–219.