corollaryroutine

Noncoordinate polynomials in every dimension at least four

So for every

n\ge4

some non-coordinate

f

has

\CC[z]/(f)\cong\CC^{[n-1]}

payoff of the main theorem

Corollary·cor:dimension·§1 Introduction and statement

Statement (paper)

For every integer

n\ge4

, there is a polynomial

f\in\CC[z_1,\ldots,z_n]

such that

\CC[z_1,\ldots,z_n]/(f)\cong\CC^{[n-1]}

and

f

is not a coordinate.

For every

n\ge4

there is a polynomial

f\in\CC[z_1,\ldots,z_n]

with

\CC[z_1,\ldots,z_n]/(f)\cong\CC^{[n-1]}

that is not a coordinate.

This extends the four-variable example of the main theorem to all

n\ge4

, so the phenomenon is not confined to one dimension. The natural route is to take the polynomial

F

from the main theorem and adjoin extra variables, which keeps the quotient a polynomial ring and keeps

f

from being a coordinate.

Classified routine:

Follows from the main theorem by adjoining

n-4

variables: the quotient remains a polynomial ring, and the critical point of

F

persists, ruling out a coordinate.

Nothing in the paper

PaperOpen

Leannot aligned

Phase 2 aligns this node to a Lean declaration.