Zariski cancellation for affine three-space

Adding an independent coordinate turns a complex line with coordinate $z$ into the plane of pairs $(z,t)$: every value of $t$ is allowed. Now take a shape described by polynomial equations over the complex numbers, of dimension three. Suppose adding one such free coordinate makes it algebraically the usual complex four-dimensional space. Must the original shape already have been the usual complex three-dimensional space? “Algebraically the same” requires coordinate rules that are polynomial in both directions, rather than just a resemblance in a drawing.[1]

Does adding one free coordinate determine whether a complex three-dimensional algebraic variety is ordinary affine three-space? The question compares polynomial coordinate structures, not just topological shapes. In algebra, start with a finitely generated commutative algebra $A$ over $\mathbb C$, with no zero divisors. It is the coordinate ring of the original irreducible affine variety.

Adjoining a free variable gives $A[t]$. Its elements are finite sums such as $a_0+a_1t+a_2t^2$, with coefficients in $A$; no equation restricts $t$. Write $R_n=\mathbb C[x_1,\ldots,x_n]$ for the polynomial ring of affine $n$-space. The exact target is: if $$A[t]\cong R_4,$$ must $$A\cong R_3?$$ Both identifications are isomorphisms of $\mathbb C$-algebras, preserving the complex constants and all sums and products. The familiar line-to-plane construction illustrates adjoining a variable; it supplies no argument that this reverse implication must hold.[1]

The complex three-dimensional case is still unsettled.[2]

A 2025 preprint reports non-affine-space constructions in dimensions at least four, a different range; contractibility alone is not the required product isomorphism.[2]

Adding a free coordinate is a simple way to enlarge a shape. Cancellation asks whether that enlargement can hide a genuinely different algebraic structure. A positive answer here would let mathematicians recognize ordinary three-coordinate space from its four-coordinate product, without first finding coordinates on the original shape.[1]

Polynomial rings are the algebraic descriptions of spaces with freely varying coordinates. Cancellation asks whether seeing such a description after one enlargement lets us recover it before enlargement. This connects factor recognition to the relation between geometry and rings of functions: losing the factor would mean that visibly ordinary polynomial behavior in the larger space conceals a different smaller variety. Requiring algebraic isomorphism is essential because topological similarity alone does not supply polynomial coordinates.[1]