Euclidean distance degree conjectures for resectioning varieties

For generic configurations of points, let RnN,h\mathcal{R}_n^{N,h} denote the corresponding resectioning variety, and let EDD\mathrm{EDD} denote Euclidean distance degree. Resectioning EDD conjectures.

EDD(Rn1,1)=3n8(n4),EDD(Rn2,2)=12n284n+147(n5),EDD(Rn2,1)=152n21152n+108(n6),EDD(Rn3,3)=883n3400n2+54563n2756(n6),EDD(Rn3,2)=803n3368n2+50683n2580(n6),EDD(Rn3,1)=212n33172n2+793n312(n8).\begin{aligned} \mathrm{EDD}(\mathcal{R}_n^{1,1})&=3n-8 && (n\ge4),\\ \mathrm{EDD}(\mathcal{R}_n^{2,2})&=12n^2-84n+147 && (n\ge5),\\ \mathrm{EDD}(\mathcal{R}_n^{2,1})&=\frac{15}{2}n^2-\frac{115}{2}n+108 && (n\ge6),\\ \mathrm{EDD}(\mathcal{R}_n^{3,3})&=\frac{88}{3}n^3-400n^2+\frac{5456}{3}n-2756 && (n\ge6),\\ \mathrm{EDD}(\mathcal{R}_n^{3,2})&=\frac{80}{3}n^3-368n^2+\frac{5068}{3}n-2580 && (n\ge6),\\ \mathrm{EDD}(\mathcal{R}_n^{3,1})&=\frac{21}{2}n^3-\frac{317}{2}n^2+793n-312 && (n\ge8). \end{aligned}

These formulas are conjectural Euclidean distance degrees for resectioning varieties. The source states that all formulas are new to the authors' knowledge except EDD(Rn3,2)\mathrm{EDD}(\mathcal{R}_n^{3,2}), attributed to an earlier conjecture.

Sources & referencesView supporting material

Primary source

Timothy Duff and Felix Rydell, “Metric Multiview Geometry – a Catalogue in Low Dimensions”, arXiv:2402.00648 (2024).

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