Counting the number of Zp- and Fp[t]-fixed points of a discrete dynamical system with applications from arithmetic statistics, III

The paper asserts, for d = p^ℓ, that the fixed-point counts mod an inert prime in number fields (Theorem 2.3), mod p in Z_p (Theorem 3.3), and mod π in F_p[t] (Theorem 5.3) are either p (for ℓ in {1,p}) or between 2 and ℓ for other ℓ, and 0 when c is not divisible by p (or π). However, over a residue field F_{p^f} the polynomial z^{p^ℓ} − z is the additive map τ − 1 with τ(x)=x^{p^ℓ}, whose image is the trace-zero subspace and whose kernel has size p^{gcd(ℓ,f)}. Thus, z^{p^ℓ} − z = −c̄ has either 0 or exactly p^{gcd(ℓ,f)} solutions, with solutions existing iff Tr_{F_{p^f}/F_{p^{gcd(ℓ,f)}}}(c̄)=0. This contradicts the paper’s Theorems 2.3, 3.3, and 5.3 whenever f>1 (e.g., K=Q(√2), p=3 inert, ℓ=2, c≡0 gives 9 solutions, not 2≤Nc(p)≤2) and even refutes Theorem 2.2/3.2 for ℓ=1 because many nonzero c̄ with trace 0 admit p solutions (contrary to ‘Nc(p)=0 for all c∉pOK’) . By contrast, the Z_p case with d=(p−1)^ℓ (Theorems 4.2/4.3) is correct: over F_p, z^{(p−1)^ℓ} equals 0 for z=0 and 1 for z≠0, giving 2/1/0 solutions for c≡0/1/−1 (mod p) . The function-field analogue (Theorems 6.2/6.3), however, is false for deg π>1; e.g., over F_{25} with d=4 and c≡0, z^4−z has 1+gcd(24,3)=4 solutions, not 2, contradicting the paper’s claim . The candidate solution’s trace-criterion for d=p^ℓ and its verification of the Z_p, d=(p−1)^ℓ case are correct and supply explicit counterexamples where the paper fails.