2026/07/22 by Sergey Sverchkov · 1 voice
#math.AC #math.AG #math.RA
We study a specific polynomial mapping G: ℂ5 → ℂ5 induced by a homogeneous polynomial F of degree 6 consisting of four tridiagonal harmonic blocks. We prove that while the Jacobian matrix of this mapping is unipotent at every point (implying det JG(x) ≡ 1), the mapping itself is not globally injective. We construct explicit algebraic sparse pairs of distinct points a ≠ b that map to the identical image G(a) = G(b) = 0. Furthermore, we perform a comprehensive classification of the zero-set structure of the corresponding gradient field, uncovering a total of 37 precise analytical complex solutions split across six distinct geometric series.