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Abhyankar-Sathaye Conjecture

The Abhyankar-Sathaye conjecture asserts in characteristic zero that a polynomial f in k[z_1,...,z_n] is a coordinate polynomial whenever

k[z_1,...,z_n]/(f)=k[t_1,...,t_(n-1)].

The isomorphism fixes the ground field. Geometrically, it asks whether every hypersurface embedding of affine (n-1)-space in affine n-space can be carried to a coordinate hyperplane by an ambient polynomial automorphism.

See also

Affine Space, Coordinate Polynomial, Hypersurface, Polynomial Ring, Stable Coordinate Conjecture, Zariski Cancellation Problem

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References

Gupta, N. "The Zariski Cancellation Problem and Related Problems in Affine Algebraic Geometry." In Proc. International Congress of Mathematicians 2022, Vol. 3: Sections 1-4 (Ed. D. Beliaev and S. Smirnov). Berlin, Germany: EMS Press, pp. 1578-1598, 2023. https://doi.org/10.4171/ICM2022/151.

Cite this as:

Weisstein, Eric W. "Abhyankar-Sathaye Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Abhyankar-SathayeConjecture.html

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