多項式の次数
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^ Mac Lane and Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)
^ Shafarevich (2003) says of a polynomial of degree zero, f ( x ) = a 0 {\displaystyle f(x)=a_{0}} : "Such a polynomial is called a constant because if we substitute different values of x in it, we always obtain the same value a 0 {\displaystyle a_{0}} ." (p. 23)
^ James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (Mechanics Magazine, Vol. LV, p. 171)
^ King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
^ 例えば以下のような用例がある:
Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27)
Childs (1995) uses ?1. (p. 233)
Childs (2009) uses ?∞ (p. 287), however he excludes zero polynomials in his Proposition 1 (p. 288) and then explains that the proposition holds for zero polynomials "with the reasonable assumption that − ∞ {\displaystyle -\infty } + m = − ∞ {\displaystyle -\infty } for m any integer or m = − ∞ {\displaystyle -\infty } ".
Axler (1997) uses ?∞. (p. 64)
Grillet (2007) says: "The degree of the zero polynomial 0 is sometimes left undefined or is variously defined as ?1 ∈ ? or as − ∞ {\displaystyle -\infty } , as long as deg 0 < deg A for all A ≠ 0." (A is a polynomial.) However, he excludes zero polynomials in his Proposition 5.3. (p. 121)
^ .mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}Barile, Margherita. "Zero Polynomial". mathworld.wolfram.com (英語).
^ Axler (1997) gives these rules and says: "The 0 polynomial is declared to have degree − ∞ {\displaystyle -\infty } so that exceptions are not needed for various reasonable results." (p. 64)
参考文献
Axler, Sheldon (1997), Linear Algebra Done Right (2nd ed.), Springer Science & Business Media, https://books.google.co.jp/books?id=ovIYVIlithQC&lpg=PA64&vq=%220+polynomial%22&pg=PA64&redir_esc=y&hl=ja#v=onepage&q=%220%20polynomial%22&f=false
Childs, Lindsay N. (1995), A Concrete Introduction to Higher Algebra (2nd ed.), Springer Science & Business Media, https://books.google.co.jp/books?id=rUApHgaTVx0C&lpg=PA279&vq=%22zero+polynomial%22&pg=PA233&redir_esc=y&hl=ja#v=onepage&q=%22zero%20polynomial%22&f=false
Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, https://books.google.co.jp/books?id=qyDAKBr_I2YC&lpg=PA288&vq=%22zero+polynomial%22&pg=PA287&redir_esc=y&hl=ja#v=onepage&q=%22zero%20polynomial%22&f=false
Grillet, Pierre Antoine (2007), Abstract Algebra (2nd ed.), Springer Science & Business Media, https://books.google.co.jp/books?id=LJtyhu8-xYwC&lpg=PA121&vq=%22degree+of+the+zero+polynomial%22&pg=PA121&redir_esc=y&hl=ja#v=onepage&q=%22degree%20of%20the%20zero%20polynomial%22&f=false
King, R. Bruce (2009), Beyond the Quartic Equation, Springer Science & Business Media, https://books.google.co.jp/books?id=9cKX_9zkeg4C&redir_esc=y&hl=ja
Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical Society, https://books.google.co.jp/books?id=L6FENd8GHIUC&lpg=PA107&vq=linear+quadratic+cubic+quartic+quintic&pg=PA107&redir_esc=y&hl=ja#v=onepage&q=linear%20quadratic%20cubic%20quartic%20quintic&f=false
Shafarevich, Igor R. (2003), Discourses on Algebra, Springer Science & Business Media, https://books.google.co.jp/books?id=hpkkJgU8rwcC&lpg=PA27&vq=%22the+degree+of+the+polynomial+is+undefined%22&pg=PA27&redir_esc=y&hl=ja#v=onepage&q=%22the%20degree%20of%20the%20polynomial%20is%20undefined%22&f=false
関連項目
ラテン語の数詞
外部リンク
Weisstein, Eric W. "Polynomial Degree". mathworld.wolfram.com (英語).
Weisstein, Eric W. "Polynomial Order". mathworld.wolfram.com (英語).
order and degree of polynomial - PlanetMath.（英語）
Definition:Degree of Polynomial at ProofWiki
表
話
編
歴
多項式
元数
多変数
次数
多項式
零多項式
定数多項式
斉次多項式
函数
次数不確定 (or −∞)（零函数）
零次（非零定数函数）
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