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Unformatted text preview: HOMEWORK #5, DUE ON 11/08 (1) The goal of this problem to formally introduce generating functions. Each student was assigned one part of this problem. If you don’t know your assignment, please let me know ASAP. Recall that ( R [[ x ]] , + , · ) is the ring of generating functions of real coeffi cients. If F ∈ R [[ x ]], denote [ x i ] F the coefficient of x i in F . Further recall that deg F = min { i : [ x i ] F negationslash = 0 } , and the main coeffient of F is [ x deg F ] F . (a) Prove that ( R [[ x ]] , + , · ) is an integral domain (you don’t need to prove that it is a ring). Definition 1. Recall that the equation F · X = G has a unique solution iff deg F ≤ deg G and F negationslash = 0 . In this case, this unique solution is called G/F . (b) Let F ∈ R [[ x ]], F negationslash = 0. Consider the equation X n = F with n even. Prove that if n negationslash  deg F or the main coefficient of F is not positive, then the equation has no solution. Also prove that if n  deg F and the main coefficient is positive, then the equation has exactly two solutions, and they are additive inverses of each other (i.e. their sum is 0). (c) Let F ∈ R [[ x ]], F negationslash = 0, and this time consider the equation X n = F with n odd. Prove that if n negationslash  deg F then the equation has no solutions....
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 Fall '09
 WILDSTROM

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