On the growth rate of polyregular functions

We consider polyregular functions, which are certain string-to-string functions that have polynomial output size. We prove that a polyregular function has output size $\mathcal O(n^k)$ if and only if it can be defined by an MSO interpretation of dimension $k$, i.e. a string-to-string transformation where every output position is interpreted, using monadic second-order logic MSO, in some $k$-tuple of input positions. We also show that this characterization does not extend to pebble transducers, another model for describing polyregular functions: we show that for every $k \in \{1,2,\ldots\}$ there is a polyregular function of quadratic output size which needs at least $k$ pebbles to be computed.