  Steady states of alternating-current (AC) circuits have been studied in
considerable detail. In 1982, Baillieul and Byrnes derived an upper bound on
the number of steady states in a loss-less AC circuit [IEEE TCAS, 29(11):
724--737] and conjectured that this bound holds for AC circuits in general. We
prove this is indeed the case, among other results, by studying a certain
multi-homogeneous structure in an algebraisation.
