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- W1486298901 abstract "The algebraic structure of the set of all proper rational vectors contained in a given rational vector space τ(s) is shown to be that of a Noetherian R <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>pr</inf> (s)-module M* (R <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>pr</inf> (s) = the ring of proper rational functions). The proper submodules M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> of M* form an ascending chain of submodules partially ordered by an invariant of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> defined as the valuation at s = ∞ of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> . The various bases of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> are examined and classified according to their property of column reduceness at s = ∞. The concept of a prime column reduced at s = basis of basis of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> is introduced. It is shown that prime bases of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> can be further classified by their MacMillan degrees and the existence of minimal MacMillan degree bases for M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> is established. A prime and minimal MacMillan degree basis of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> extends Forney's concept of a minimal polynomial basis of τ(s) for the R <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>pr</inf> (s)-module M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> . The MacMillan degrees of the columns of such bases form a set of invariants for M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> which are defined as the generalized invariant dynamical indices of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> , and a simple relation is established between (i) the generalized invariant dynamical indices of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> , (ii) the orders of zeros at s = ∞ of M <inf xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</inf> , and (iii) the Forney invariant dynamical indices of τ(s)." @default.
- W1486298901 created "2016-06-24" @default.
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- W1486298901 date "1983-06-01" @default.
- W1486298901 modified "2023-09-25" @default.
- W1486298901 title "Proper, Minimal Macmillan Degree, Bases of Rational Vector Spaces" @default.
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- W1486298901 doi "https://doi.org/10.23919/acc.1983.4788293" @default.
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