. We give function algebraic characterizations of the polynomial time computable functions using unbounded recursion schemes without introducing extra notions such as safe and normal in variables and tiering in word algebras. Key words. polytime functions, function algebras, recursion schemes. Subject classications. 68Q15. 1. Introduction Cobham (1965) characterized the class Fptime of polynomial time computable functions as the smallest class of functions containing certain initial functions, and closed under composition and the following variant of primitive recursion: the function f is dened by bounded recursion on notation (brn) from g; h 0 ; h 1 ; k if f(0; ~y) = g(~y); f(s 0 (x); ~y) = h 0 (x; ~y; f(x; ~y)); (if x 6= 0) f(s 1 (x); ~y) = h 1 (x; ~y; f(x; ~y)) provided that f(x; ~y) k(x; ~y) for all x; ~y, where s 0 and s 1 are the binary successor functions with s 0 (x) = 2 x and s 1 (x) = 2 x + 1. Although Cobham's characterization has been fruitful and yielded a ..