The Abhyankar-Sathaye Conjecture via Boundary Reduction and Affine Collapse

This preprint proves the Abhyankar-Sathaye conjecture over an algebraically closed field of characteristic zero. It shows that every surjective polynomial map f: A³ₖ→A¹ₖ whose generic fiber is isomorphic to A²ₖ is a coordinate polynomial. The argument proceeds through a shell-adapted compactification, completed-local boundary analysis at the bad fibers, construction of a rank-two translation lattice of vertical locally nilpotent derivations, and an affine collapse criterion. A key local step is a codimension-one boundary-purity statement excluding extra vertical height-one phenomena, which is then combined with reflexive extension and formal glueing to obtain the global coordinate conclusion. The manuscript is self-contained and is intended as a research preprint in affine algebraic geometry.