  Angluin et al. proved that population protocols compute exactly the
predicates definable in Presburger arithmetic (PA), the first-order theory of
addition. As part of this result, they presented a procedure that translates
any formula $\varphi$ of quantifier-free PA with remainder predicates (which
has the same expressive power as full PA) into a population protocol with
$2^{O(\text{poly}(|\varphi|))}$ states that computes $\varphi$. More precisely,
the number of states of the protocol is exponential in both the bit length of
the largest coefficient in the formula, and the number of nodes of its syntax
tree.
  In this paper, we prove that every formula $\varphi$ of quantifier-free PA
with remainder predicates is computable by a leaderless population protocol
with $O(\text{poly}(|\varphi|))$ states. Our proof is based on several new
constructions, which may be of independent interest. Given a formula $\varphi$
of quantifier-free PA with remainder predicates, a first construction produces
a succinct protocol (with $O(|\varphi|^3)$ leaders) that computes $\varphi$;
this completes the work initiated in [STACS'18], where we constructed such
protocols for a fragment of PA. For large enough inputs, we can get rid of
these leaders. If the input is not large enough, then it is small, and we
design another construction producing a succinct protocol with one leader that
computes $\varphi$. Our last construction gets rid of this leader for small
inputs.
