Polynomials in a Single Ordinal Variable

We denote by ON the class of all ordinals, and define a polynomial P in the single ordinal variable 5 to be an expression of the form ~ { X ~ ~ ( ~ ) C ~ ( P ) ; 5 < Z(P)}, where z(P) is an ordinal known as the length of P, each e&P) is an ordinal, the (1 + 5)th exponent of P, and each ce(P) is a positive ordinal, the (1 + 6)th coefficient of P. For any ordinal a , we give the obvious interpretation to the expression P(a). Let P, Q be two polynomials; we say that P and Q are equal (P = Q ) if they are equal as formal expressions, i.e. if Z(P) = Z(Q), e&P) = er(Q) and cr(P) = c&Q) for each 6 < l(P). On the other hand, we say that P and Q are equivalent ( P = Q ) if P(a) = &(a) €or each a E ON. We will have occasion to extend this latter concept by saying that P and Q are equivalent over a nonempty subclass r of ON (P I r = Q I T ) if P(a) = Q(a) for all a E T. Addition of polynomials is defined in the obvious formal manner: given two polynoniials P,Q, we define P + Q t o be the polynomial R such that Z(R) = l (P) + Z(Q), e&R) = ee(P) for < Z(P) and e,(,,,&R) = e&Q) for 6 < Z(Q), etc. If P , Q are polynomials such that P = R + Q + S for some polynomials R , S , then we call Q a segment of P. and say that Q is initial (final) if R = Z (S = Z), where 2 is the unique polynomial of length 0. Incidentally, we adopt the convention that Z(a) = 0 for all a. If Q, R are both initial segments of some polynomial P, then we shall say that Q is shorter than R if z ( Q ) < Let P be any polynomial, and let a, /3 be any ordinals; we define the polynomial PpZa to be P if P(@) < a , and to be the shortest initial segment Q of P for which Q(8) 2 a otherwise. It is clear that Pp;& is well-defined. A polynomial P is called homogeneous if cc(P) = 1 for all 5 < Z(P). Since the righthand distributive law holds for ordinals, it is easy to see that each polynomial is equivalent to some homogeneous polynomial. Let a polynomial P be given. We define an ordinal-valued function hp of two variables by taking hp(x, y ) to be the unique ordinal 16 for which xyb 2 P(x) < xu(@ + 1) . Furthermore, we define the parameter e(P) to be sup (ee(P); 5 < Z(P)), and write hp(x) for hp(x, e(P)). We observe that hp(z) 2 1 for every polynomial P Theorem 1. Let P be any homogeneous polynomial, and take any x, y E ON with x 5 y . TIierL hr(y) 5 h,,(x).