A new Hopf algebra of formal multiple polylogarithms is defined for every field, with Hodge and motivic realizations, proposed as the conjectural Hopf algebra of framed mixed Tate motives.
On the Goncharov Depth Conjecture and polylogarithms of depth two
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abstract
We prove the surjectivity part of Goncharov's depth conjecture. We also show that the depth conjecture implies that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function $\mathrm{Li}_{n-d+1,1,\dots,1}(a_1,a_2,\dots,a_d)$, and we prove this latter statement for $d=2$.
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The Hopf algebra of formal multiple polylogarithms
A new Hopf algebra of formal multiple polylogarithms is defined for every field, with Hodge and motivic realizations, proposed as the conjectural Hopf algebra of framed mixed Tate motives.