Hi Myriam, Thanks for bringing this up as it points out an update we need to make. The latest review of the genus Persicaria and Polygonum can be viewed at The Catalog of Life (CoL) site. Here's Persicaria:
http://www.catalogueoflife.org...
Persicaria microcephala is noted as a synonym of Polygonum microcephalum as of 12-Oct-12, CoL China.
http://www.catalogueoflife.org...
The Plant list references Tropicos where Polygonum microcephalum var. microcephalum is noted as the accepted name as of 2003.
http://www.tropicos.org/Name/5...
Persicaria runcinata per CoL is a synonym of Polygonum runcinatum as of 12-Oct-2012.
http://www.catalogueoflife.org...
Per Tropicos, a synonym of Polygonum runcinatum var. runcinatum as of 2003.
http://www.tropicos.org/Name/5... I'll update this one per the Col since it is the latest taxonomic scrutiny.
I imagine it will take a while for many sites to update the names, that is unless they get revised again.
Hope this helps.