On the Goncharov depth conjecture and polylogarithms of depth two

We prove the surjectivity part of Goncharov’s depth conjecture over a quadratically closed field. We also show that the depth conjecture implies that multiple polylogarithms of depth d and weight n can be expressed via a single function \({{\,\textrm{Li}\,}}_{n-d+1,1,\dots ,1}(a_1,a_2,\dots ,a_d)\) ${\text{Li}}_{n-d+1,1,\cdots ,1}(a_1,a_2,\cdots ,a_d)$ , and we prove this latter statement for \(d=2\) $d=2$ .