  Characteristic p multizeta values were initially studied by Thakur, who
defined them as analogues of classical multiple zeta values of Euler. In the
present paper we establish an effective criterion for Eulerian multizeta
values, which characterizes when a multizeta value is a rational multiple of a
power of the Carlitz period. The resulting "t-motivic" algorithm can tell
whether any given multizeta value is Eulerian or not. We also prove that if
zeta_A(s_1,...,s_r) is Eulerian, then zeta_A(s_2,...,s_r) has to be Eulerian.
When r=2, this was conjectured (and later on conjectured for arbitrary r) by
Lara Rodriguez and Thakur for the zeta-like case from numerical data. Our
methods apply equally well to values of Carlitz multiple polylogarithms at
algebraic points and zeta-like multizeta values.
