Mahler's theorem

In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses continuous p-adic functions in terms of polynomials. Over any field, one has the following result:

Let ${\displaystyle (\Delta f)(x)=f(x+1)-f(x)}$ be the forward difference operator. Then for polynomial functions f we have the Newton series

${\displaystyle f(x)=\sum _{k=0}^{\infty }(\Delta ^{k}f)(0){x \choose k},}$

where

${\displaystyle {x \choose k}={\frac {x(x-1)(x-2)\cdots (x-k+1)}{k!}}}$

is the kth binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function f is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity. Mahler's theorem states that if f is a continuous p-adic-valued function on the p-adic integers then the same identity holds. The relationship between the operator Δ and this polynomial sequence is much like that between differentiation and the sequence whose kth term is x^k.

It is remarkable that as weak an assumption as continuity is enough; by contrast, Newton series on the field of complex numbers are far more tightly constrained, and require Carlson's theorem to hold. It is a fact of algebra that if f is a polynomial function with coefficients in any field of characteristic 0, the same identity holds where the sum has finitely many terms.