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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Linearised polynomial ： ウィキペディア英語版 Linearised polynomial In mathematics, a linearised polynomial (or ''q''- polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of ''q'' and the coefficients come from some extension field of the finite field of order ''q''. We write a typical example as :$L(x)\; =\; \backslash sum\_^n\; a\_i\; x^,\; \backslash text\; a\_i\; \backslash text\; F\_\; (\backslash text\; GF(q^m))\; \backslash textm.$ This special class of polynomials is important from both a theoretical and an applications viewpoint. The highly structured nature of their roots makes these roots easy to determine. ==Properties== * The map ''x'' → ''L''(''x'') is a linear map over any field containing F''q'' * The set of roots of ''L'' is an F''q''-vector space and is closed under the ''q''-Frobenius map * Conversely, if ''U'' is any F''q''-linear subspace of some finite field containing F''q'', then the polynomial that vanishes exactly on ''U'' is a linearised polynomial. * The set of linearised polynomials over a given field is closed under addition and composition of polynomials. * If ''L'' is a nonzero linearised polynomial over $F\_$ with all of its roots lying in the field $F\_$ an extension field of $F\_$, then each root of ''L'' has the same multiplicity, which is either 1, or a positive power of ''q''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Linearised polynomial」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.