The embedding conjecture for quasi-ordinary hypersurfaces

TL;DR

This paper extends the theory of Abhyankar-Moh to quasi-ordinary polynomials and proves the embedding conjecture for this class, showing that isomorphic hypersurfaces are equivalent to coordinates.

Contribution

It generalizes the Abhyankar-Moh theory to quasi-ordinary polynomials and proves the embedding conjecture for this class using approximate roots and Newton polygons.

Findings

Proved the embedding conjecture for quasi-ordinary hypersurfaces.

Extended Abhyankar-Moh theory to a broader class of polynomials.

Established criteria for hypersurface equivalence in affine space.

Abstract

This paper has two objectives: we first generalize the theory of Abhyankar-Moh to quasi-ordinary polynomials, then we use the notion of approximate roots and that of generalized Newton polygons in order to prove the embedding conjecture for this class of polynomials. This conjecture -made by S.S. Abhyankar and A. Sathaye- says that if a hypersurface of the affine space is isomorphic to a coordinate, then it is equivalent to it.