The multiplicative Euclidean distance degree formula for parameterized hypersurface classes

Let Xm,ndX_{m,n}^d be a variety parameterizing a set of hypersurfaces with the same Euclidean distance degree, let HH be a point of Xm,ndX_{m,n}^d, and let kk be the number of parameters in the parametrization of the hypersurface class. The multiplicative EDD formula. The conjecture states

EDdegree(Xm,nd)=EDdegree(H)mkEDdegree(Xk,nd).EDdegree(X_{m,n}^d)=EDdegree(H)^{m-k}EDdegree(X_{k,n}^d).

The formula is proposed for classes whose members have the same Euclidean distance degree, but the supplied context gives no proof or resolution. Its general status therefore remains open.

Sources & referencesView supporting material

Primary source

Oliver Gäfvert, “Computational complexity of learning algebraic varieties”, arXiv:1910.03305 (2020).

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