B.4 Polynomial roots under coefficient perturbation
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Komorebi P. — 8 September 2026
This block formalizes square root uniqueness and nearby-root stability for finite systems of integer polynomials. The polynomials and their partial derivatives are evaluated directly. No root-uniqueness or perturbation-stability premise is assumed.
B.4.1 Notation
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Let \(p\) be prime, \(L\ge1\), and
Write
For an integer \(c\), the perturbed system is the actual polynomial list
There is no restriction on the sign of \(c\), and \(c=0\) is allowed. Divisibility conclusions give lower bounds on precision and allow zero differences.
B.4.2 1. Unique roots for square systems
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If \(d=e\), \(A_x\) is surjective, and \(F(x)\equiv0\pmod{p^L}\), then
The stronger comparison lemma proves that any two exact roots of \(F\) in \(B_L(x)\) coincide, without needing a separate lower-root hypothesis.
Proof implemented. Map every arbitrary p-adic root into the actual residue rings \(\mathbb Z/p^n\mathbb Z\), and choose its canonical integer representative. Evaluation commutes with reduction, so these representatives are genuine finite polynomial roots. The preceding count theorem gives exactly \(p^{d-d}=1\) next-level residue lift. Inductively any two roots agree at every precision. The p-adic residue maps separate points, so the roots are equal. Existence is the preceding checked tower theorem.
This extends uniqueness from the limit of a selected sequence to all exact roots in the prescribed ball.
B.4.3 2. Coefficient perturbations preserve smooth liftability
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If \(p^L\mid c\), the actual identities give
Thus if \(F(x)\equiv0\pmod{p^L}\) and \(A_x\) is surjective, then
This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition \(G_i=F_i+C(c)H_i\), rather than supplied as hypotheses. Algebraic congruence helpers also permit composite moduli; exact p-adic root existence uses prime \(p\).
B.4.4 3. Nearby roots for rectangular systems
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Let \(M\ge1\) and \(p^M\mid c\). Suppose \(A_x\) is surjective and \(a\in\mathbb Z_p^d\) is an exact root of \(F\) with \(a\equiv x\pmod p\). Then
No uniqueness is claimed when \(d>e\).
Proof implemented. Truncate \(a\) to an integer vector \(y\) modulo \(p^M\). It is an actual lower root of \(F\), and its reduced Jacobian agrees with \(A_x\). Apply the coefficient-perturbed existence theorem at precision \(M\), then combine the two congruences through \(y\).
B.4.5 4. Full precision for the unique square root
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Suppose \(d=e\), \(M\ge L\ge1\), \(p^M\mid c\), and \(A_x\) is surjective. If
then
In particular, for \(G=F+p^MH\), no p-adic digits below depth \(M\) are changed in the selected root.
The formalization also proves a combined existence theorem. Starting only with \(F(x)\equiv0\pmod{p^L}\), it constructs an original root \(a\in B_L(x)\) and proves that there is exactly one \(b\in B_L(x)\) satisfying \(G(b)=0\) and \(b\equiv a\pmod{p^M}\). The comparison theorem shows that the precision bound holds for every root of \(G\) in that ball, so uniqueness is not obtained merely by adding a restrictive displacement condition.
Proof implemented. Section 3 produces a nearby root \(b'\). Since \(M\ge L\), it belongs to \(B_L(x)\). The perturbed Jacobian remains surjective, and square uniqueness identifies \(b'\) with the arbitrary root \(b\).