We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a nonsurjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through t...

then almost all series (1) diverge almost everywhere on I zl = 1 . Here almost all refers to the set of t (in the usual Lebesgue sense), while almost everywhere refers to the set of z on the circumference of the unit circle (again in the usual sense) . Only recently [1] was it observed that some nontrivial interesting assertions similar to the above with almost everywhere replaced by everywhere...

The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R((G≤0)) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n...

We describe a new algorithm for computing exp f where f is a power series in CJxK. If M(n) denotes the cost of multiplying polynomials of degree n, the new algorithm costs (2.1666 . . . + o(1))M(n) to compute exp f to order n. This improves on the previous best result, namely (2.333 . . . + o(1))M(n). The author recently gave new algorithms for computing the square root and reciprocal of power ...

2.1 Series over a graded monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Graded monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Topology on K〈〈M〉〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Distance on K〈〈M〉〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

An algorithm is introduced and shown to lead to a unique infinite product representation for a given formal power series A(z) with A(O)= 1. The infinite product is 1; n (1 + b,z'n)> " =I

In order to analyze the singularities of a power series P (t) on the boundary of its convergent disc, we introduced the space Ω(P ) of opposite power series in the opposite variable s=1/t, where P (t) was, mainly, the growth function (Poincaré series) for a finitely generated group or a monoid [S1]. In the present paper, forgetting about that geometric or combinatorial background, we study the ...