claimroutinestructure_uncertain
Cusp and translation identities
Shows and for the translation
identities behind the base algebra and the critical point
Claim·§3 The affine-space fibre and the critical point·inside thmNoncoordinate polynomial with affine three-space zero fibre
Statement (paper)
Direct expansion of gives . Moreover, for formal , the translation , satisfies , since the terms from the square and cube cancel.
In plain English
Expanding the definitions of gives . For formal , the translation , gives .
Why it's here
This is the algebraic bookkeeping behind the proof of the main theorem. The cusp relation and the translation, where the terms cancel, let the later claims identify the base algebra with a polynomial ring in and derive the ambient identity that yields the critical point.
Classified routine:
These identities are direct polynomial expansions of the paper’s definitions and the binomial formulas; the cancellation of the s^6 terms requires no new technique.