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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 Polynomial Diophantine equation ： ウィキペディア英語版 Polynomial Diophantine equation In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers. (In another usage ) ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made initial studies of integer Diophantine equations. An important type of polynomial Diophantine equations takes the form: :$sa+tb=c\backslash$ where ''a'', ''b'', and ''c'' are known polynomials, and we wish to solve for ''s'' and ''t''. A simple example (and a solution) is: :$s(x^2+1)+t(x^3+1)=2x\backslash$ :$s=-x^3-x^2+x\backslash$ :$t=x^2+x.\backslash$ A necessary and sufficient condition for a polynomial Diophantine equation to have a solution is for ''c'' to be a multiple of the GCD of ''a'' and ''b''. In the example above, the GCD of ''a'' and ''b'' was 1, so solutions would exist for any value of c. Solutions to polynomial Diophantine equations are not unique. Any multiple of $ab$ (say $rab$) can be used to transform $s$ and $t$ into another solution $s\text{'}=s+rb$ $t\text{'}=t-ra$: :$(s+rb)a+(t-ra)b=c.\backslash$ Some polynomial Diophantine equations can be solved using the extended Euclidean algorithm, which works as well with polynomials as it does with integers. ==References== * 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「Polynomial Diophantine equation」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.