汉森·穆伦猜想
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发布时间：2024-04-03 05:45
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In mathematics, Mahler's theorem, named after Kurt Mahler (1903–1988), identifies one of various respects in which analysis is simpler with p-adic numbers than with real numbers.
In any field, one has the following result. Let
(\Delta f)(x)=f(x+1)-f(x)\,
be the forward difference operator. Then for polynomial functions f we have the Newton series:
f(x)=\sum_{k=0}^\infty (\Delta^k f)(0){x \choose k},
where
{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 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.
热心网友
时间：2024-04-12 07:26
热心网友
时间：2024-04-12 07:27
Author(s): Shuqin Fan; Wenbao Han
Journal: Proc. Amer. Math. Soc. 132 (2004), 15-31.
MSC (2000): Primary 11T55, 11F85, 11L40, 11L07
Posted: May 8, 2003
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Abstract: In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.
