Mahler's theorem
Encyclopedia
In mathematics, Mahler's theorem, introduced by , expresses continuous p-adic functions in terms of polynomials.
In any field
, one has the following result. Let
be the forward difference operator. Then for polynomial functions f we have the Newton series:
where
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 xk.
It is remarkable that as weak an assumption as continuity is enough; by contrast, Newton series on the complex number field 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.
In any field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, one has the following result. Let
be the forward difference operator. Then for polynomial functions f we have the Newton series:
where
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
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
.
Mahler's theorem states that if f is a continuous p-adic
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
-valued function on the p-adic integers then the same identity holds.
The relationship between the operator Δ and this polynomial sequence
Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
is much like that between differentiation and the sequence whose kth term is xk.
It is remarkable that as weak an assumption as continuity is enough; by contrast, Newton series on the complex number field are far more tightly constrained, and require Carlson's theorem
Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem about a summable expansion of an analytic function. It is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for expansions in other bases of...
to hold.
It is a fact of algebra that if f is a polynomial function with coefficients in any field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0, the same identity holds where the sum has finitely many terms.