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Theorem · exact classification

FPRD-LL-T03

Classification of two-state loops on polynomial graphs

Exact statement

For deg⁡f≥2\deg f\ge2, the graph y=f(x)y=f(x) admits a nontrivial rational two-variable homogeneous linear loop if and only if f(x)=αxm+βxf(x)=\alpha x^m+\beta x for some m≥2m\ge2, α≠0\alpha\ne0, and β∈Q\beta\in\mathbb Q.

StatusProved by polynomial identity and degree comparison
External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

A two-state update has x'=ax+by and y'=cx+dy. Invariance on infinitely many graph points forces f(ax+bf(x))=cx+df(x) as a polynomial identity.

Hypotheses and scope

  • The ground field is Q.
  • The update is homogeneous.
  • The visible orbit is infinite.

Proof or evidence

Degree m² versus m forces b=0. Coefficient comparison then rules out a constant term and more than one nonlinear exponent unless a is a root of unity, which would make the orbit finite. The displayed triangular update proves sufficiency.

Two-state classification →

Verification notes

The b≠0 degree argument, constant term, multiple nonlinear exponents, and all rational roots-of-unity cases were checked.

Limitations

  • Affine updates can be homogenized with an extra constant coordinate and are not classified by this two-state theorem.

Open work

Determine the corresponding classification for affine two-variable updates and for larger ground fields.