Steven Charlton

An explicit quadruple ratio

In his celebrated proof of Zagier's polylogarithm conjecture for weight 3 Goncharov introduced a "triple ratio", a projective
invariant akin to the classical cross-ratio. He has also conjectured the existence of "higher ratios" that should play an important role for Zagier's conjecture in higher weights. Recently, Goncharov and Rudenko proved the weight 4 case of Zagier's conjecture with a somewhat indirect method where they avoided the need to define a corresponding "quadruple ratio". We propose an explicit candidate for such "quadruple ratio" and as a by-product we get an explicit formula for the Borel regulator of K_7 in terms of the tetralogarithm function (joint work with H. Gangl and D. Radchenko).