Cycles of rational mappings in algebraically closed fields of positive characteristics
Abstract
Let K be a field. Let us define K̅ as a formal set K∪{∞} (which can be identified with P1(K)). For convenience we put that the degree of a zero polynomial is zero (not -∞). For relatively prime polynomials f,g∈K[X] of degrees n, m and leading coefficients an, bm, respectively, we define rational function ϕ(X) = f(X)/g(X) as a mapping ϕ: K̅ ↦ K̅ as follows:
f(ξ)/g(ξ) for ξ∈K, g(ξ)≠0
∞ for ξ∈K, g(ξ)=0
ϕ(ξ)= an/bm for ξ=∞, n=m
∞ for ξ=∞, n>m
0 for ξ=∞, n<m,
where we put 1/0 as ∞.
More generally for (f,g) ≠ (0,0) and ϕ = f/g we put ϕ = f1/g1, where f1 = f/d, g1 = g/d, d = gcd(f,g).
A k-tuple x0, ...,xk-1 of distinct elements of K̅ is called a cycle of ϕ of length k if
ϕ(xi) = xi+1 for i = 0, 1, ..., k-2 and ϕ(xk-1) = x0.
The set of all positive integers which are not lengths of a cycle for ϕ will be denoted by Exc(ϕ).
The case of a rational mapping not of the form (aX + b)/(cX + d) over algebraically closed field of characteristic zero was solved by I.N. Baker [2], who showed that Exc(ϕ) is always finite and gave all possible examples of Exc(ϕ). However for positive characteristic the situation differs, in fact it was shown in [3] that for some polynomials ϕ the set Exc(ϕ) is infinite.
The aim of this paper is to prove that for a large class of rational ϕ over algebraically closed field of positive characteristic the set Exc(ϕ) is "lacunar", i.e. either Exc(ϕ) is finite or Exc(ϕ) = {a1 < a2 < ...} with ai+1/ai > λ > 1 for all i.
Keywords
rational functions; lacunar sets
References
2. I.N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc., 39 (1964), 615-622.
3. T. Pezda, Cycles of polynomials in algebraically closed fields of positive characteristic, Colloq. Math., 67 (1994), 187-195.
4. T. Pezda, Cycles of polynomials in algebraically closed fields of positive characteristic, II, Colloq. Math., 71 (1996), 23-30.
Instytut Matematyki, Uniwersytet Wrocławski Poland
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