Moving-Root Tails and Exact Low-Cost Path Words for Primitive Binary Quadratic Germs
Year: 2026
Type: preprint
Abstract: Let $R=\kk[[x,y]]$, where $\kk$ is algebraically closed of characteristic zero, and let $q=a u^2+2buv+c v^2\in R\otimes_\kk \Sym^2(V^*)$ be primitive with nonzero discriminant $b^2-ac$. Iterated point blowups of the coefficient ideal produce proper and infinitely-near primitive centers. At every center we label the new exceptional component by $N$, $B$, or $M$ according as the initial binary quadratic pencil is nonbranch, constant-root branch, or moving-root branch, and we charge the square of the common coefficient order. We first prove that the multiplicity of a primitive successor is bounded by the vanishing order of the scalar gcd of the current initial triple. The inequality is strict whenever a nonbranch exceptional is followed by a branch exceptional. At multiplicity one, an inherited rank-one tangent direction forces every branch child to have the same constant root and every nonbranch child to be terminal. For moving-root exceptionals we factor the homogeneous initial triple as $H(x,y)(P(x,y)u+Q(x,y)v)^2,$ $\qquad \mu=\deg H+2\deg[P:Q].$ Under path square cost at most $12$, a moving-root vertex is either a terminal $M_2$, or an $M_3$ with one simple successor followed by at most two same-label $B_1$ vertices and a terminal $N_1$. Combining these local restrictions gives an exact classification of maximal path words after forgetting multiplicities, proximity, and most root labels. Precisely six families occur: $N^r,$ $\quad N^rB^sN,$ $\quad B^rN^s,$ $\quad B^rM,$ $\quad MB^rN,$ $\quad BNB^rN,$ with explicit sharp numerical ranges. Every one of the $47$ allowed word instances is realized by a primitive polynomial germ. The strict nonbranch-to-branch inequality also yields a short proof that every low-cost constant-root return has multiplicity pattern $(2,2,1,1)$; this proof is independent of the normal-form and degree-threshold arguments in our earlier work. We do not classify proximity-decorated whole trees or all multiplicity-decorated realizability types.
Authors: Ueoka, Yoshiki, Nagi, Akari, Sui
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172294259
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