$\operatorname{circPoly}$ is nonzero on affinely independent quadruples

Proved

NearEnemy.circPoly_ne_zero_of_affineIndependent

by mysticflounder · Sep 18, 2026 · Mathlib 0df444a (Lean v4.33.1)

circumsphere-polynomialgeneral-positionnear-enemypolynomial-method

Let $a,b,c,e$ be points in EuclideanSpace ℝ ι that are affinely independent (the hypothesis hind : AffineIndependent ℝ ![a,b,c,e]). Then the circumsphere-detecting polynomial is nonzero on this quadruple:

\operatorname{circPoly}(a,b,c,e) \neq 0.

Affine independence of the four points — full $3$ -dimensional span of the difference vectors — is enough to keep circPoly from vanishing. Only this direction is proved here; the converse, that vanishing of circPoly forces a cospherical or degenerate quadruple, is not part of this development. Note that affine independence of four points requires ambient dimension at least $3$ , so in the plane the hypothesis has no instances. This is the highest-codimension nonvanishing input to the polynomial method that builds a generic projection avoiding cospherical quadruples; the coplanar and no-three-collinear cases are handled by their own lemmas.

Preamble

import Mathlib
import Definitions.Def_NearEnemyDefs

universe u_1
open scoped RealInnerProductSpace
open scoped Classical
open MvPolynomial
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
variable {ι : Type*} [Fintype ι]
open NearEnemy

Formal statement

theorem NearEnemy.circPoly_ne_zero_of_affineIndependent {a b c e : EuclideanSpace ℝ ι}
    (hind : AffineIndependent ℝ ![a, b, c, e]) :
    circPoly a b c e ≠ 0 := by sorry

Source

Prior art: Lund-Sheffer-de Zeeuw, Bisector energy and few distinct distances, SoCG 2015, LIPIcs vol. 34, 537-552, DOI 10.4230/LIPIcs.SOCG.2015.537, footnote 1 on p. 538, state that E(P) = 2n(n-1) when every pair of distinct points has a distinct perpendicular bisector, with the count of trivial quadruples that proves the floor (this footnote is not in arXiv:1411.6868v1); the asymptotic floor E(P) = Omega(n^2) is in their section 3.4. The generic planar projection that is injective, keeps general position and transports distances is Erdos-Furedi-Pach-Ruzsa, The grid revisited, Discrete Math. 111 (1993), proof of Theorem 3.1. Formalized in https://github.com/mysticflounder/lean-formalizations/blob/dd46c17a2a034d7bfa0df02e7f77834d35592864/lean/LeanFormalizations/Geometry/Euclidean/NearEnemyTheorem.lean#L1170-L1240

View graph

circPoly⁡\operatorname{circPoly}circPoly is nonzero on affinely independent quadruples

$\operatorname{circPoly}$ is nonzero on affinely independent quadruples