Random polynomials on local surfaces: rationality and reciprocity in every residue characteristic

The problem, standards and verification were set and directed by Asvin G; Claude (Anthropic) and GPT-6-Astra (OpenAI) produced the text and proofs, with the Lean formalization cited in [Lean].

A monic polynomial over a complete discrete valuation ring defines a horizontal curve on an arithmetic surface. Fixing its initial homogeneous form turns one point blowup into independent branch problems with an exact change of volume. The resulting finite recursion computes the joint law of ramification indices and residue degrees as rational functions of the residue-field size, the same over every complete DVR with finite residue field. Arbitrary vertical units remain in the geometry, so the recursion applies in every characteristic and at every prime. Hensel lifting assembles these local computations into polynomial probabilities. The quadratic exponents in the transition weights form a product in noncommuting variables; inverting the residue-size variable reverses that product and proves the two factorization symmetries conjectured by Bhargava–Cremona–Fisher–Gajović. Extracting coefficients gives a totally ramified divisor recursion and the root-moment, split and unramified-field formulas of Bhargava–Cremona–Fisher–Gajović, Buhler–Goldstein–Moews–Rosenberg and Shmueli.

Are a random polynomial's factorization probabilities determined by the residue-field size, or do they retain finer information about the local field?

When every ramification index is prime to $p$, extensions admit descriptions using roots of unity and roots of a uniformizer [Yi22, Theorem 3.6]. Del Corso–Dvornicich proved effective rationality for tame monic types [DCD00]. Yin treated all three models by integrating root counts over extensions and recursively computing discriminant valuations from Teichmüller digits (over $\mathbb F_Q$, the residue lifts satisfying $a^Q=a$): equal leading digits group conjugates, unequal digits contribute to the discriminant, and subtracting a common digit reduces the problem [Yi22, Section 1.1]. Fixing the base's absolute ramification index and residue degree gives rationality in $q$ and projective-density asymptotics [Yi22, Theorems 3.1–3.2]. The auxiliary densities (the other terms of the recursion) need not be symmetric [GWY, Introduction].

For a Galois cover with group $G$, G, Wei and Yin assume an equivariant resolution hypothesis (for relevant $g\in G$, a $g$-equivariant resolution with normal-crossings ramification divisor). Poincaré duality makes paired densities (sums for Frobenius and inverse Frobenius at fixed inertia) palindromic forms of weight $w$ (functions $\nu$ satisfying $\nu(-m)=q^{-wm}\nu(m)$), where $m$ is residue-extension degree; individual densities need not be palindromic [GWY, Theorem 3.10].

A polynomial type sums densities over admissible pairs (subgroups $H\subset G$ with cosets $gH=Hg$), up to conjugacy [GWY, Lemma 4.4]. These are inversion-closed: inverting Frobenius preserves the inertia and decomposition orbits. Pairing therefore gives palindromy for each type. Normalization by $|\mathbb P^n(\mathbb F_q)|$ and tame rationality give projective reciprocity [GWY, Theorem 4.5]; the monic/cluster identity remains open there [GWY, p. 18].

Yin's root-of-uniformizer parameterization requires tameness [Yi22, Theorem 3.6]; its conjugate-difference valuations change at wild primes. Write $\zeta_m$ for a primitive $m$th root of unity and normalize $v_p(p)=1$. His example $L=\mathbb Q_p(\zeta_{p^3-1},p^{1/2})$ has ramification index two and inertia degree three. For a nonzero Teichmüller digit $a_1$ with $a_1^{p^3}=a_1$, the valuation step is $v_p(2a_1p^{1/2})=v_p(2)+1/2$, equal to $1/2$ only when $p\ne2$ [Yi22, Section 1.1].

GWY's duality requires an equivariant resolution over the integer ring [GWY, Hypothesis 3.3]. The arXiv versions of [GWY] stated the projective result at every prime (Theorem 5.10) by resolving the resultant locus, whose equivariant splitting of formal neighbourhoods into “tangential and normal formal discs” needs the residue characteristic prime to the isotropy groups, hence $p>n$. The corrected version [GWY2] withdraws the wild claim and proves the tame case only; the wild statement remains unproven, not known to be false.

At wild primes, direct computation still settles small degrees: BCFG prove the conjecture in degrees two and three [BCFG22, p. 8], including the primes excluded in Weiss's formulas [We13, Theorems 4.1 and 4.3]. In arbitrary degree, restricting the type can simplify the recursion: Shmueli's single unramified field is governed by a divisor recurrence over $\mathbb Z_p$ [Sh22, Theorem 2]. Complete splitting instead separates by residue classes; BGMR's recursion uses Gaussian weights $q^{-\binom{d+1}{2}}$ for degrees $d\ge0$, over every complete DVR with finite residue field [BGMR06, Theorem 1.1, Remark 1.2, Lemma 5.1(4)], the weights in (1.7).

These recursions handle restricted types at every prime; general uniformity treats broader problems but leaves finitely many primes aside [DL01, Theorem 0.2; CL08, Remark 16.4.2], and BCFG suggest a connection to Denef–Loeser and Pas [Pa89] only for sufficiently large primes. A computation for every type must continue through wild ramification without changing its rules.

One cluster computation therefore also supplies the monic law by rational-function substitution, without assigning a probability meaning to a residue size below one.

1. Exact blowup volumes give one finite recursion over every complete DVR with finite residue field (Theorems 1.1 and 6.2).
2. Comparing assembly and blowup equations proves both BCFG symmetries (Theorems 1.2 and 9.3).
3. Selecting types consisting of one field gives a divisor recursion governed by the Euclidean algorithm (Theorem 10.2).
4. Specializing type variables recovers the root-count, split and unramified-field recursions (Section 11).
5. Lean 4 kernel checking certifies rationality and reciprocity using only the three axioms listed in Remark 1.3.

Hensel factorization separates residue roots. A root of multiplicity $c$, translated to zero, gives a monic Weierstrass factor $f\equiv x^c\pmod\pi$. Work on a complete regular local $O$-algebra $(R,\mathfrak m,k)$ of dimension two with $O/\pi O\simeq R/\mathfrak m$; initially $R=O[[x]]$, $\mathfrak m=(\pi,x)$. A curve is a principal ideal $(f)\subset R$; its field factors give the type. The boundary is $V(\pi)$, the union of vertical components. For a component $t=0$, contact means $\operatorname{length}R/(f,t)$.

For $R/(f)$ finite free over $O$, put $N=\operatorname{rank}_O R/(f)$ and choose $b_i\bmod f$ as an integral basis. On the ideals lifting $(f\bmod\pi)$, intrinsic measure means the pushforward of product Haar measure on $(\pi O)^N$ by $z\mapsto(f+\sum_i z_i b_i)$. Write $\mu_R$ for this measure; Proposition 3.2 proves independence of the choices.

Blowing up replaces the closed point by $E=\mathbb P(\mathfrak m/\mathfrak m^2)$, the line of directions. Put $d=\operatorname{ord}_{\mathfrak m}f=\max\{j:f\in\mathfrak m^j\}$ and $\operatorname{in}_d f=f\bmod\mathfrak m^{d+1}\in\mathfrak m^d/\mathfrak m^{d+1}$. Let the tangent divisor be $D=\operatorname{div}_E(\operatorname{in}_d f)=\sum_P r_PP$, with multiplicities $r_P=\operatorname{mult}_P(D)$ and $\delta_P=[k(P):k]$. A cell fixes $d,D$ and the boundary contacts. At $P$, put $R_P:=\widehat{\mathcal O}_{\operatorname{Bl}_{\mathfrak m}(\operatorname{Spec}R),P}$. For a local exceptional equation $e_P$, the child ideal is $(f'_P)$ with $f'_P:=f/e_P^d$. Write $\mu_{\mathrm{parent}}:=\mu_R$, $\mu_P:=\mu_{R_P}$. Gluing these independent child ideals recovers one parent ideal, with unnormalized measure

The blowup of a cluster need not leave another cluster. Writing $t=\pi$, the chart $x=tz$ puts a finite tangent direction on the single vertical component $t=0$. The other chart has $t=xw$, hence $\pi=xw$: at $x=w=0$ the exceptional line meets the old vertical component. Following a branch therefore requires two kinds of state. Each retains the component multiplicities and the curve's contacts; the surface and its vertical unit remain arbitrary.

Proof plan for Lemma 2.1.

1. Present the special fibre and cancel prime parameters to compute its curve length.
2. Lift a residue basis one $\pi$-adic digit at a time; regularity cancels every putative relation.

Proof. The complete local ring $R/\pi R$ is a $k$-algebra, hence has characteristic $p$. Hensel's lemma lifts each residue element uniquely to a root of $z^q-z$: the derivative is congruent to $-1$ modulo the maximal ideal (indeed it equals $-1$ here). Characteristic $p$ makes these roots a field under addition and multiplication. They give a coefficient field. Sending $T,X$ to $t,x$ now gives the asserted surjection. Its associated graded map is an isomorphism: $\operatorname{gr}_{\mathfrak m}R=k[T,X]$, and the initial ideal of $(\pi)$ is generated by the initial monomial of $\pi$, since initial forms multiply in this domain. Successive approximation in the complete local rings makes the original map an isomorphism.

Lift $\bar f$ to $F\in k[[T,X]]$. Finite contact says that none of the axes in the vertical monomial divides $F$. Repeated cancellation of those prime parameters shows that $\bar f$ is a nonzerodivisor. In the auxiliary curve ring $A_F=k[[T,X]]/(F)$, whose quotients $A_F/(T^aX^b)$ compute special-fibre lengths, the same parameters are nonzerodivisors. Filtering by their powers gives $\operatorname{length}A_F/(T^a)=ac$ and, in the crossing case, the exact sequence

Consequently $\dim_k R/(\pi,f)=N$.

If $\pi v=fw$ in $R$, reduction modulo $\pi$ forces $w=\pi w_1$, and cancellation gives $v=fw_1$. Thus $\pi$ is regular on $A_f$. Choose $b_1,\ldots,b_N\in R$ lifting a basis modulo $(\pi,f)$. Successive subtraction of residue coefficients writes any class in $A_f$ as $\sum_i z_i b_i$, with $z_i\in O$: the errors lie in $\pi^jA_f$ and tend to zero $\mathfrak m$-adically. The quotient is complete because ideals in a complete noetherian local ring are closed. A relation among the $b_i$ has every coefficient divisible by $\pi$; regularity allows cancellation and repetition. Its coefficients lie in every $\pi^jO$, hence vanish. This proves freeness. ∎

Blowing up the closed point preserves this class of surfaces. On its two charts substitute $x=tz$ or $t=xw$. The exceptional equation and a parameter on the exceptional line form regular parameters at each completed local ring. At a closed point $P$ of degree $\delta$, let $O_\delta$ be the integer ring of the unramified extension $K_\delta/K$ of degree $\delta$, with a specified identification of its residue field with $k(P)$. Such an extension is unique up to unique isomorphism respecting that identification [Se79]. A monic lift $\widetilde g$ of the minimal polynomial of a generator of $k(P)/k$ presents $O_\delta=O[X]/(\widetilde g)$. Its simple residue root lifts by Hensel's lemma in the completed local ring at $P$, giving an embedding of $O_\delta$: the kernel is zero because $O_\delta$ is a DVR and its uniformizer $\pi$ remains nonzero. The substituted equation for $\pi$ is still a unit times a monomial. In the crossing case the two old components meet the exceptional line at distinct rational points, so at most two vertical components pass through any child point. Each child is treated from here on with $O_\delta$ as its base ring and residue size $q^\delta$.

Write $k_\delta=O_\delta/\pi O_\delta$ for the residue field of $O_\delta$.

On a reduced generic fibre, the normalization of $A_f$ is the product of the integer rings of its field factors, whether or not those fields are separable over $K$. A branch valuation is the normalized valuation $v:L^\times\to\mathbb Z$ of a field factor $L$ of $A_f[1/\pi]$, with $v(L^\times)=\mathbb Z$. The vertical equation gives $v(\pi)=a v(t)$ or $a v(t)+b v(x)$. Its residue degree is the degree of its residue field over $k$. These valuations read the type; measuring the curves realizing each branch gives its probability. For related valuation methods, see [FGMN15, GMN12, OW20].

Only $S(a;1)$ is the base case of Theorem 6.2; the crossing contact-one laws are computed by (6.7) and agree with Lemma 2.2.

Haar measure on equations would count unit multiples separately. Infinitesimally, multiples of $f$ do not change its ideal, leaving the quotient $R/(f)$, free of rank $N$, as the tangent lattice. An integral basis suggests $N$ coordinates. To give those coordinates a measure, every lift of a fixed residue ideal must have a unique representative $f+\sum_i z_i b_i$, with $z_i\in\pi O$, and changes of basis or generator must preserve Haar measure. In the quadratic cluster the proposed coordinates are just the two coefficients.

For $0\ne h\in R[1/\pi]$, set $|h|:=q^{-\max\{j\in\mathbb Z:h\in\pi^jR\}}$ and $|0|=0$.

The ring $R$ is complete also for the $\pi$-adic topology: a $\pi$-adic Cauchy sequence converges $\mathfrak m$-adically, and the closed ideals $\pi^jR$ preserve its congruences in the limit. It is separated since $\pi^jR\subseteq\mathfrak m^j$. Thus $\mathscr R=R[1/\pi]$, with norm given by divisibility by $\pi$ and unit ball $R$, is a nonarchimedean Banach $K$-algebra. Cauchy sequences have bounded denominators, reducing completeness to that of $R$.

Uniqueness in (3.2) removes the unit freedom within a chart; comparing charts still requires control of their Jacobians.

Normalizing a whole state requires counting its residue ideals. A residue generator restricted to $T=0$ is a power series in $X$ of order $c$, with $c$ zero coefficients followed by a nonzero one; dividing generator counts by unit orbits gives ideal counts.

The child measures define a product measure. Identifying it with a parent event requires every allowed tuple of child ideals to come from exactly one parent ideal.

A tuple of child ideals supplies no choice of generators that could be glued into a parent equation. On each finite thickening of the exceptional line $E$, their quotients have disjoint supports, so the kernel of the direct-sum quotient map glues the ideals. The glued divisor must still descend to one equation. Its restriction to $E$ has tangent degree $d$, while $E$ restricts with degree $-1$; hence adding $dE$ gives degree zero. Trivializing this line bundle recovers a parent equation. The cohomology governing that trivialization also measures the index between parent and product child orders.