Sparse-degree construction for polynomial graphs

Exact statement

For $f(x)=\sum_k f_kx^k\in\mathbb Q[x]$ of degree at least two, the graph $y=f(x)$ has a nontrivial rational linear loop with $r_f=|\{1\}\cup\{k:f_k\ne0\}|$ state variables. Hence $r_f\le\deg f+1$ , and $r_f\le\deg f$ when $f(0)=0$ .

StatusProved by a diagonal exponential construction

External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

The occupied exponents of f determine a finite shift basis for the scaling orbit x(n)=2^n, y(n)=f(2^n).

Hypotheses and scope

f has rational coefficients and degree at least two.
Updates are homogeneous and rational.

Proof or evidence

The sequences 2^{kn} for distinct occupied exponents are Vandermonde-independent and shift diagonally with rational eigenvalues 2^k. Taking x and y as the first two coordinates yields the loop.

Sparse-degree construction →

Verification notes

The constant, linear, and nonlinear monomial cases and the state count were checked separately.

Limitations

The construction gives an upper bound for the graph; it does not prove that every orbit on the graph needs that many states.

Open work

Extend the construction from polynomial graphs to rational parametrized curves and higher-dimensional varieties.