The ancestor weight-drop conjecture for ϕ4\phi^4 periods

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Let GG be a primitive log-divergent ϕ4\phi^4 graph with loop number hGh_G and completion G‾\overline{G}, and suppose P(G)∈ZN[ξN]P(G)\in\mathcal{Z}_N[\xi_N] for some NN. Let anc⁡(G‾)\operatorname{anc}(\overline{G}) be the ancestor obtained from the completion by a maximal sequence of product and double-triangle reductions. Ancestor weight-drop conjecture. The weight drop of anc⁡(G‾)\operatorname{anc}(\overline{G}) exists and equals the weight drop of P(G)P(G), namely 2hG−32h_G-3 minus the maximum weight of P(G)P(G).

The claim concerns periods known to be multiple polylogarithms at roots of unity. The weight filtration exists more generally, but the source reports supporting data only in this restricted setting; it remains open.

References

Primary source

Erik Panzer and Oliver Schnetz, “The Galois coaction on ϕ^4 periods”, arXiv:1603.04289 (2017).

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