# The formal inverse and its tree formula

- kind: theorem
- status: proved (evidence tier V3)
- year: 1982
- authors: H. Bass, E. H. Connell and D. Wright (1982); D. Wright (1987, 1989); the Abhyankar–Gurjar inversion formula
- in one line: The inverse of x − H is a sum over rooted trees; invertible ⇔ the sums stop

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

## Relations

- related to [jc-n](/wiki/Statement:jc-n.md) — 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.

## References

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](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.

## Relations to this page

- used by [zhao-vanishing](/wiki/Statement:zhao-vanishing.md) — So the map has a polynomial inverse exactly when those terms vanish for large m, which is a statement about the formal inverse.

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