# Abhyankar's two points at infinity

- kind: theorem
- status: proved (evidence tier V3)
- year: 1977
- authors: S. S. Abhyankar (Tata lectures, notes by Balwant Singh)
- in one line: Each component of a Jacobian pair has at most two points at infinity

**Statement.** Let $\operatorname{char}k=0$. If $J(f,g)\in k^{\times}$, then over $\bar k$ the top-degree form of $f$ (and of $g$) is a product of powers of at most two pairwise coprime linear forms.

A polynomial has $r$ points at infinity when its top-degree form factors, over the algebraic closure, into powers of $r$ distinct linear forms. Abhyankar proved that each component of a Jacobian pair has at most two (Cor. 18.15 of his Tata lectures). The same bound holds for any grading with positive weights. He also proved that the components of an automorphism have one point at infinity, and that a pair with one point at infinity and degree above 2 can be lowered in degree by an automorphism (Thm 19.2).[1]

Together these reduce the plane conjecture to one configuration. If JC₂ fails, a counterexample of least degree has both components with exactly two points at infinity. The same lectures prove JC₂ equivalent to three statements about all Jacobian pairs (Thm 19.4): every component has one point at infinity; its Newton polygon is a triangle; or one component degree divides the other. So [the plane conjecture is equivalent to the claim that no Jacobian pair has a component with exactly two points at infinity](/wiki/Statement:jc-plane.md).

This is the setting of the Newton-polygon attack, including the [gcd bounds](/wiki/Statement:plane-gcd-bounds.md). McKay and Wang reproved and refined the two-point statement through Newton-polygon edges (Proc. AMS 1991).

## Relations

- related to [jc-plane](/wiki/Statement:jc-plane.md) — So the plane conjecture is equivalent to the claim that no Jacobian pair has a component with exactly two points at infinity.

## References

1. S. S. Abhyankar, *Lectures on Expansion Techniques in Algebraic Geometry*, notes by Balwant Singh, Tata Institute Lectures 57 (1977), searched through the [archive.org OCR index](https://archive.org/details/Balwant_Singh___Lectures_on_Expansion_Techniques_In_Algebraic_Geometry).

## Relations to this page

- used by [plane-gcd-bounds](/wiki/Statement:plane-gcd-bounds.md) — The same Żołądek paper proves the plane conjecture for every Keller map of topological degree at most 5, the wall drawn as rows 3–5 on the JC₂ chart. Newton-polygon methods from Abhyankar's two-points-at-infinity framework underlie the gcd arguments.

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