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Generic 2:2 relation for natural parameters of degenerate tetrahedra

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Let α,β,γ,δ,ς∈R\alpha,\beta,\gamma,\delta,\varsigma\in\mathbb R satisfy

∣α∣+∣β∣+∣γ∣+∣δ∣>∣α+β+γ+δ∣,αβγδ>0,|\alpha|+|\beta|+|\gamma|+|\delta|>|\alpha+\beta+\gamma+\delta|,\qquad \alpha\beta\gamma\delta>0,

and let ϱ+(α,β,γ,δ)\varrho_+(\alpha,\beta,\gamma,\delta) denote the unique positive root of the discriminant quadratic described in the surrounding discussion. Generic 2:2-relation conjecture. The equations referenced in the source generically define a 2 ⁣: ⁣22\!:\!2 relation between the natural parameters of degenerate tetrahedra and the semi-algebraic set

{α,β,γ,δ,ς∈R∣∣α∣+∣β∣+∣γ∣+∣δ∣>∣α+β+γ+δ∣, αβγδ>0, ς>ϱ+(α,β,γ,δ)}.\left\{\alpha,\beta,\gamma,\delta,\varsigma\in\mathbb R\mathrel{\bigm|}|\alpha|+|\beta|+|\gamma|+|\delta|>|\alpha+\beta+\gamma+\delta|,\ \alpha\beta\gamma\delta>0,\ \varsigma>\sqrt{\varrho_+(\alpha,\beta,\gamma,\delta)}\right\}.

The conjecture describes the generic multiplicity of the parameter correspondence in the region where the preceding analysis predicts three real roots.

References

Primary source

Timothy F. Havel, “An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros”, arXiv:2204.08089 (2025).

algebraic-geometrysemi-algebraic-setstetrahedra
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