Mahler's theorem [SEP] named after
Mahler's theorem
In mathematics, Mahler's theorem, introduced by , expresses continuous "p"-adic functions in terms of polynomials. Over any field, one has the following result:
Let formula_1 be the forward difference operator. Then for polynomial functions "f" we have the Newton series
where
is the "k"th 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
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 proved the following result:
Mahler's theorem: If "f" is a continuous p-adic-valued function on the "p"-adic integers then the same identity holds.
Important results Hensel's lemma.
Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular