The computational content of atomic polymorphism

We show that the number-theoretic functions definable in the atomic polymorphic system (${\mathbf{F}}_{\mathbf{at}}$) are exactly the extended polynomials. Two proofs of the above result are presented: one, reducing the functions’ definability problem in ${\mathbf{F}}_{\mathbf{at}}$ to definability in the simply typed lambda calculus ($\lambda ^{\rightarrow }$) and the other, directly adapting Helmut Schwichtenberg’s strategy for definability in $\lambda ^{\rightarrow }$ to the atomic polymorphic setting. The uniformity granted in the polymorphic system, when compared with the simply typed lambda calculus, is emphasized.