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The effective potentials for massless scalar and vector quantum fieldtheories on D dimensional manifolds with p compact noncommutative extradimensions are evaluated by means of dimensional regularization implemented byzeta function techniques. It is found that the zeta function associated withthe one-loop operator may not be regular at the origin. Thus, the relatedheat-kernel trace has a logarithmic term in the short t asymptotics expansion.Consequences of this fact are briefly discussed.Comment: 9 pages, Latex AMS style, typos corrected, numerical coefficients  corrected, references adde

ONE-LOOP EFFECTIVE POTENTIAL FOR SCALAR AND VECTOR FIELDS ON HIGHER-DIMENSIONAL NONCOMMUTATIVE FLAT MANIFOLDS