Problem 6. Let No [×]] be the set of all nonzero generating functions Σ0 anx" with coefficients in No (having at least one nonzero coefficient), and observe that No [x] is closed under the standard multiplication of generating functions: for any f(x) = Σx=0 anx" and g(x) : Σno bx in No [x], f(x)g(x) = ( anx n=0 bnxn ∞ = n ") ( Ĉ bx^" ) := Σ ( ^ ak bn-k) 2” € No[2]. n=0 n=0 k=0 • A generating function f(x) = No[[x] \ {1} is called indecomposable if whenever the equality f(x) = g(x)h(x) holds for some g(x), h(x) = No [x], either g(x) = 1 or h(x) = 1. • A generating function s(x) = No [x] is called supported if either s(x) = 1 or s(x) can be written as a product of finitely many indecomposable generating functions in No [x] (repetitions of factors are allowed). (a) Find a generating function in No [x] that is not supported. (b) Is it possible to find, for each f(x) = No [×], a supported generating function s(x) = No [x] such that s(x) f(x) is also supported? (c) Is it possible to find a supported generating function s(x) = No [x] such that s(x) f(x) is a supported generating function for all f(x) = No [x]?
Problem 6. Let No [×]] be the set of all nonzero generating functions Σ0 anx" with coefficients in No (having at least one nonzero coefficient), and observe that No [x] is closed under the standard multiplication of generating functions: for any f(x) = Σx=0 anx" and g(x) : Σno bx in No [x], f(x)g(x) = ( anx n=0 bnxn ∞ = n ") ( Ĉ bx^" ) := Σ ( ^ ak bn-k) 2” € No[2]. n=0 n=0 k=0 • A generating function f(x) = No[[x] \ {1} is called indecomposable if whenever the equality f(x) = g(x)h(x) holds for some g(x), h(x) = No [x], either g(x) = 1 or h(x) = 1. • A generating function s(x) = No [x] is called supported if either s(x) = 1 or s(x) can be written as a product of finitely many indecomposable generating functions in No [x] (repetitions of factors are allowed). (a) Find a generating function in No [x] that is not supported. (b) Is it possible to find, for each f(x) = No [×], a supported generating function s(x) = No [x] such that s(x) f(x) is also supported? (c) Is it possible to find a supported generating function s(x) = No [x] such that s(x) f(x) is a supported generating function for all f(x) = No [x]?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.1: The Geometry And Algebra Of Vectors
Problem 24EQ
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![Problem 6. Let No [×]] be the set of all nonzero generating functions Σ0 anx" with coefficients in No
(having at least one nonzero coefficient), and observe that No [x] is closed under the standard multiplication
of generating functions: for any f(x) = Σx=0 anx" and g(x) : Σno bx in No [x],
f(x)g(x) = (
anx
n=0
bnxn
∞
=
n
") ( Ĉ bx^" ) := Σ ( ^ ak bn-k) 2” € No[2].
n=0
n=0 k=0
• A generating function f(x) = No[[x] \ {1} is called indecomposable if whenever the equality f(x) =
g(x)h(x) holds for some g(x), h(x) = No [x], either g(x) = 1 or h(x) = 1.
• A generating function s(x) = No [x] is called supported if either s(x) = 1 or s(x) can be written as
a product of finitely many indecomposable generating functions in No [x] (repetitions of factors are
allowed).
(a) Find a generating function in No [x] that is not supported.
(b) Is it possible to find, for each f(x) = No [×], a supported generating function s(x) = No [x] such that
s(x) f(x) is also supported?
(c) Is it possible to find a supported generating function s(x) = No [x] such that s(x) f(x) is a supported
generating function for all f(x) = No [x]?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bf8e00f-8bd7-492b-bde4-883297a247cf%2F1c52cdc2-bda7-4e80-bd76-36b02150fe25%2Fajb7uv7_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 6. Let No [×]] be the set of all nonzero generating functions Σ0 anx" with coefficients in No
(having at least one nonzero coefficient), and observe that No [x] is closed under the standard multiplication
of generating functions: for any f(x) = Σx=0 anx" and g(x) : Σno bx in No [x],
f(x)g(x) = (
anx
n=0
bnxn
∞
=
n
") ( Ĉ bx^" ) := Σ ( ^ ak bn-k) 2” € No[2].
n=0
n=0 k=0
• A generating function f(x) = No[[x] \ {1} is called indecomposable if whenever the equality f(x) =
g(x)h(x) holds for some g(x), h(x) = No [x], either g(x) = 1 or h(x) = 1.
• A generating function s(x) = No [x] is called supported if either s(x) = 1 or s(x) can be written as
a product of finitely many indecomposable generating functions in No [x] (repetitions of factors are
allowed).
(a) Find a generating function in No [x] that is not supported.
(b) Is it possible to find, for each f(x) = No [×], a supported generating function s(x) = No [x] such that
s(x) f(x) is also supported?
(c) Is it possible to find a supported generating function s(x) = No [x] such that s(x) f(x) is a supported
generating function for all f(x) = No [x]?
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