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Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ( a t ) - h D ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number.
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This content will become publicly available on December 28, 2025
Hilbert polynomials for finitary matroids
We consider a tuple Φ = (φ_1,...,φ_m) of commuting maps on a finitary matroid X. We show that if φ satisfies certain conditions, then for any finite set A⊆X, the rank of {φ_1^{r_1}···φ_m^{r_m}(a): a ∈ A and r_1+···+r_m = t} is eventually a polynomial in t (we also give a multivariate version of the polynomial). This allows us to easily recover Khovanskii's theorem on the growth of sumsets, the existence of the classical Hilbert polynomial, and the existence of the Kolchin polynomial. We also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as a new result on the growth of Betti numbers in simplicial complexes.
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- Award ID(s):
- 2103240
- PAR ID:
- 10644257
- Publisher / Repository:
- MSP (Mathematical Sciences Publishers)
- Date Published:
- Journal Name:
- Pacific Journal of Mathematics
- Volume:
- 333
- Issue:
- 2
- ISSN:
- 0030-8730
- Page Range / eLocation ID:
- 273 to 308
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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