Polynomial time deterministic identity testingalgorithm for ${\mathrm{\Sigma }}^{[3]}\mathrm{\Pi }\mathrm{\Sigma }{\mathrm{\Pi }}^{[2]}$ circuits via Edelstein-Kelly type theorem for quadratic polynomials

In this work we resolve conjectures of Beecken, Mitmann and Saxena [BMS13] and Gupta [Gup14], by proving an analog of a theorem of Edelstein and Kelly for quadratic polynomials. As immediate corollary we obtain the first deterministic polynomial time black-box algorithm for testing zeroness of ${\mathrm{\Sigma }}^{[3]}\mathrm{\Pi }\mathrm{\Sigma }{\mathrm{\Pi }}^{[2]}$ circuits.