We study size bounds for conjunctive query (CQ) results which in recent years have played a crucial role in database theory. In particular, we compare the so-called entropic bound which is known to be asymptotically tight but not known to be computable, and its computable relaxation called the polymatroid bound, which is generally not tight. We focus here on conjunctive queries under acyclic functional dependencies. These queries are known to be well-behaved in the sense that, in the case without projections, both bounds coincide. We show that this picture changes when projections are allowed: in this case, even for acyclic functional dependencies, there is in general a polynomial gap between the two bounds. We complement this negative result by showing a special case for which the polymatroid bound is tight under acyclic functional dependencies and projections that is characterized by the position of the output variables in a topological order of the query variables.

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