Functional Composition Patterns and Power Series Reversion
نویسندگان
چکیده
منابع مشابه
A fast algorithm for reversion of power series
We give an algorithm for reversion of formal power series, based on an efficient way to evaluate the Lagrange inversion formula. Our algorithm requires O(n1/2(M(n)+MM(n1/2))) operations where M(n) and MM(n) are the costs of polynomial and matrix multiplication respectively. This matches an algorithm of Brent and Kung, but we achieve a constant factor speedup whose magnitude depends on the polyn...
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Our main result is elementary and concerns the relationship between the multiplicative groups of the coordinate and endomorphism rings of the formal additive group over a field of characteristic p > 0. The proof involves the combinatorics of base p representations of positive integers in a striking way. We apply the main result to construct a canonical quotient of the module of Coleman power se...
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Given a formal power series g(x) = b0 +b1x+b2x2 +··· and a nonunit f(x) = a1x+ a2x2+··· , it is well known that the composition of g with f , g(f(x)), is a formal power series. If the formal power series f above is not a nonunit, that is, the constant term of f is not zero, the existence of the composition g(f(x)) has been an open problem for many years. The recent development investigated the ...
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Let f and g be two convergent power series in R[[z]] or C[[z]], whose first n terms are given numerically with a λ n-bit precision for a fixed constant λ> 0. Assuming that g0=0, we will show in this paper that the first n coefficients of f ◦ g can be computed with a λn-bit precision in time Õ(n). Using Newton iteration, a similar complexity bound holds for power series reversion of g. Our metho...
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Our main result is elementary and concerns the relationship between the multiplicative groups of the coordinate and endomorphism rings of the formal additive group over a field of characteristic p > 0. The proof involves the combinatorics of base p representations of positive integers in a striking way. We apply the main result to construct a canonical quotient of the module of Coleman power se...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1960
ISSN: 0002-9947
DOI: 10.2307/1993433