Local height arguments toward the dynamical Mordell-Lang conjecture

TL;DR

This paper proves the dynamical Mordell-Lang conjecture for certain regular endomorphisms of complex affine space with a degree gap, showing that all periodic curves are vertical lines under specific conditions.

Contribution

It establishes a new criterion involving degree gaps that ensures the conjecture holds and characterizes periodic curves as vertical lines for a broad class of endomorphisms.

Findings

Proves the conjecture for endomorphisms with degree gap condition.

Shows all periodic curves are vertical lines under the condition.

Provides examples demonstrating the condition's optimality.

Abstract

We consider regular endomorphisms of the complex affine space with a degree gap $k$. They are endomorphisms $f$ of $\mathbb{A}_{\mathbb{C}}^{N}$ of the form $f(x_1,\dots,x_N)=(f_1(x_1,\dots,x_N)+g_1(x_1,\dots,x_N),\dots,f_N(x_1,\dots,x_N)+g_N(x_1,\dots,x_N))$, in which $f_1,\dots,f_N$ are homogeneous polynomials of degree $d$ with no nonzero common zeros and $g_1,\dots,g_N$ are polynomials of degree $\leq d-k$. Such an endomorphism extends to an endomorphism of $\mathbb{P}_{\mathbb{C}}^{N}$. Let $H_{\infty}=\mathbb{P}_{\mathbb{C}}^{N}\setminus\mathbb{A}_{\mathbb{C}}^{N}$ be the infinity hyperplane and we denote $f_{\infty}$ as the induced endomorphism of $H_{\infty}$. Suppose that $k$ is twice greater than the multiplicities of $f_{\infty}$ at the periodic closed points, i.e. $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Then we prove that $f$ satisfies the dynamical Mordell-Lang conjecture for curves. As a by-product of our proof, we show that in this case every periodic curve of $f$ is a "vertical line", i.e. a straight line passing through the origin. There are many examples which satisfy our condition $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Indeed, we prove that for every $d\geq2$, a general endomorphism $f_{\infty}$ of $H_{\infty}\cong\mathbb{P}_{\mathbb{C}}^{N-1}$ of degree $d$ satisfies $\max\limits_{P\in H_{\infty}(\mathbb{C})}e_{f_{\infty}}(P)\leq(N-1)!\cdot2^{N-1}$. So if we take $k=(N-1)!\cdot2^N+1$, then $f$ will satisfy our condition if $f_{\infty}$ is general (of an arbitrary degree $d\geq k$). Moreover, we provide examples to illustrate that this condition is optimal to force every periodic curve to be a vertical line, in the sense that one cannot change "$>$" into "$\geq$".