Star Transforms Without Type 2 Singular Directions and Conflitti's Conjecture on Elementary Symmetric Polynomials

The star transform integrates a function on the plane along m rays with a common vertex, with directions γ_1, …, γ_m and nonzero weights c_1, …, c_m. In the inversion formula of Ambartsoumian and Latifi, the directions ψ at which Σ_j c_j Π_{i≠j} ⟨ψ, γ_i⟩ vanishes, called singular directions of Type 2, cause instability. They showed that every star with an even number of rays has such directions, and that for m = 3 every choice of weights admits ray directions without them. For odd m ≥ 5 this was left as a conjecture, which the Oberwolfach Report 21/2023 records as unproven. We prove the conjecture for every odd m and all nonzero real weights. The proof reduces the problem, by nearly parallel rays, to a rational function of one variable without real zeros, which we build by nesting explicit three-pole clusters. The same report recalls a conjecture of Conflitti (2006): for even r, the real zero set of the elementary symmetric polynomial e_r in n variables contains no linear subspace of dimension r. Previously the cases r = 2 (Conflitti) and n = r + 1 (Ambartsoumian and Latifi; this case also follows from the irreducibility of e_{n−1}) were known. We prove the conjecture for all even r and all n, using Descartes' rule of signs and a parity argument. Hence the largest subspace in the zero set has dimension min(n, r − 1). For even r ≥ 4 we also show that the only (r − 1)-dimensional subspaces in the zero set are coordinate subspaces. Exact computer certificates for explicit stars with at most 21 rays illustrate the construction; the proofs do not depend on them. This is an unrefereed note.