Conjectured initial values of the ML-degree polynomial

From papers

Let ϕ(n,d)\phi(n,d) denote the polynomial that gives the dimension of the space of quadrics in nn variables passing through dd general points and tangent to (n+12)d1\binom{n+1}{2}-d-1 general hyperplanes. Initial-value conjecture.

ϕ(0,d)=(1)d1,ϕ(1,d)=(2)d1.\phi(0,d)=(-1)^{d-1},\qquad \phi(-1,d)=(-2)^{d-1}.

These formulas extend the enumerative-geometric polynomial ϕ(n,d)\phi(n,d) to the initial nonpositive values n=0n=0 and n=1n=-1, following the program of combinatorial reciprocity. The source notes that these formulas were previously conjectured; their resolution is not established in the supplied context.

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Sources & referencesView supporting material

Primary source

Maciej Gałązka, “Initial values of ML-degree polynomials”, arXiv:2212.12588 (2024).

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