Logarithmic source curves in polynomial fiber products

Abstract

Let $k$ be a field of characteristic zero and let $f,g\in k[x]$ be nonconstant. We study rational lifts of $f(a)$ through $g$ that do not arise from a composition $f=g\circ h$. To each non-graph component of $f(X)=g(Y)$ we attach its logarithmic source curve, namely the smooth compactification of its normalization with reduced boundary. The main geometric result is a sharp contact formula at infinity: if $N=\deg g/\gcd(\deg f,\deg g)$, then every one-infinity non-graph source has $X$-degree $N$, and in general the $X$-degree is $N$ times the number of boundary points. Over number fields this yields a finite symmetric-difference expansion of $S$-integral new lifts. Active one-infinity sources give exactly the power terms in height counts; positive-rank admissible two-infinity sources give logarithmic $S$-unit families; and inactive one-infinity sources, rank-zero two-infinity sources, and the remaining components contribute only finitely many inputs. Primitive one-infinity source classes have only polylogarithmic overlap, and ordered configuration covers introduce no new exponent. Over $\mathbb Q$, the $B^{1/2}$ boundary is precisely the quadratic Bilu--Tichy source cell.