Componentwise height bounds for polynomial value-set lifting

TL;DR

This paper establishes componentwise height bounds for polynomial value-set lifting over number fields, analyzing the contribution of rational components with various geometric properties.

Contribution

It provides sharp height bounds for polynomial components, characterizes when square-root growth occurs, and links involutive symmetries to specific polynomial structures.

Findings

Componentwise height bounds depend on geometric properties of polynomial components.

Square-root growth occurs precisely from active rational components with specific infinity points and degree.

Every square-root source implies the existence of an involutive affine symmetry in the polynomial.

Abstract

Let $f,g \in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\circ h$. The main theorem gives componentwise height bounds. For a rational component of $f(X)-g(Y)=0$ with one geometric point at infinity and projection degree $d_X(C)$ to the $X$-line, the corresponding contribution has the sharp power-log order $B^{[k:\mathbb{Q}]/d_X(C)}(\log B)^{q_{k,S}}$, where $q_{k,S}=\mathrm{rk}\,\mathcal{O}_{k,S}^{\ast}=|S|-1$, precisely when its $X$-parametrization is $S$-active. Rational components with two geometric points at infinity contribute only polylogarithmically, and all other components contribute finitely many inputs. Over $\mathbb{Q}$, square-root growth after graph removal occurs exactly from active rational components with one geometric point at infinity and $d_X(C)=2$. We give an explicit thin exceptional set for the generic multiplicity theorem and prove that every square-root source forces $g$ to have an involutive affine symmetry.