2015/09/22 by Joseph Malkoun, Malkoun, Joseph
Mathematics · #15A15 #20B30 #30C10 #74H05 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #math.DG #math.MG #msc:15A15 #msc:20B30 #msc:30C10 #msc:74H05
paper · pdf · doi:10.48550/arxiv.1509.06629
8 page. Added two constructions for polynomials associated to $n$ distinct points on the Riemann sphere
openalex publication_date 2015/09/22 · arxiv created 2015/11/20 · arxiv updated 2015/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Denoting by Cn(X) the configuration space of n distinct points in X, with X being either Euclidean 3-space 𝔼3 or hyperbolic 3-space ℍ3 or ℂP1 , by \mathscrPk,d the vector space of homogeneous complex polynomials in the variables z0, …, zk of degree d, and by Obsnd the set of all d-subsets of \1,…,n\, the symmetric group Σn acts on Cn(ℝ3) by permuting the n points and also acts in a natural way on Obsnd. With n = k+d, the space \mathscrPk,d has dimension \binomnd, which is also the number of elements in Obsnd. It is thus natural to ask the following question. Is there a family of continuous maps fI: Cn(X) → ℙ\mathscrPk,d, for I ∈ Obsnd (here ℙ is complex projectivization), which satisfies fI(σ.x) = fσ.I(x), for all σ∈ Σn and all x ∈ Cn(X), and such that, for each x ∈ Cn(X), the polynomials fI(x), for I∈ Obsnd, each defined up to a scalar factor, are linearly independent over ℂ? We provide two closely related smooth candidates for such maps for each of the two cases, Euclidean and hyperbolic, which would be solutions to the above problem provided a linear independence conjecture holds. Our maps are natural extensions of the Atiyah-Sutcliffe maps. Moreover, we get two constructions of actual solutions of the above problem for X = ℂP1, as we prove linear independence for these last two constructions. These last two constructions are classical in character, and can be viewed as higher dimensional versions of Lagrange polynomial interpolation. They appear to be new.