. . . . . "O anel de polin\u00F4mios com coeficientes em um anel qualquer e qualquer n\u00FAmero de indeterminadas \u00E9 a generaliza\u00E7\u00E3o dos an\u00E9is como , dos polin\u00F4mios com coeficientes reais p(x) = a0 + a1 x + ... + an xn. De forma gen\u00E9rica, para definir-se o anel dos polin\u00F4mios precisa-se: \n* um anel A dos coeficientes; \n* um conjunto S das indeterminadas. As indeterminadas aqui tem um significado puramente abstrato, n\u00E3o sendo exigido que S tenha nenhuma estrutura. Assim, \u00E9 conveniente que S seja um conjunto de s\u00EDmbolos, e (para evitar ambiguidades) que seja disjunto de A. ,[1]"@pt . . . . . . . . . . "En matem\u00E0tiques, especialment en el camp de l'\u00E0lgebra abstracta, un anell de polinomis o \u00E0lgebra de polinomis \u00E9s un anell (que tamb\u00E9 \u00E9s una \u00E0lgebra commutativa) format a partir del conjunt de polinomis en una o m\u00E9s variables (o ) amb coeficients en un altre anell, sovint un cos. Els anells de polinomis s'han tractat bastament en diversos \u00E0mbits de les matem\u00E0tiques, com per exemple al , a la construcci\u00F3 de cossos de descomposici\u00F3, o a la comprensi\u00F3 del funcionament dels operadors lineals. Moltes conjectures importants tenen a veure amb anells de polinomis, com la , i han influ\u00EFt l'estudi d'altres tipus d'anells, com els o els anells de s\u00E8ries formals de pot\u00E8ncies. Un concepte relacionat \u00E9s el d' sobre un espai vectorial."@ca . "Polynomring"@de . "\u5728\u62BD\u8C61\u4EE3\u6578\u4E2D\uFF0C\u591A\u9805\u5F0F\u74B0\u63A8\u5EE3\u4E86\u521D\u7B49\u6578\u5B78\u4E2D\u7684\u591A\u9805\u5F0F\u3002\u4E00\u500B\u74B0 \u4E0A\u7684\u591A\u9805\u5F0F\u74B0\u662F\u7531\u4FC2\u6578\u5728 \u4E2D\u7684\u591A\u9805\u5F0F\u69CB\u6210\u7684\u74B0\uFF0C\u5176\u4E2D\u7684\u4EE3\u6578\u904B\u7B97\u7531\u591A\u9805\u5F0F\u7684\u4E58\u6CD5\u8207\u52A0\u6CD5\u5B9A\u7FA9\u3002\u5728\u7BC4\u7587\u8AD6\u7684\u8A9E\u8A00\u4E2D\uFF0C\u7576 \u70BA\u4EA4\u63DB\u74B0\u6642\uFF0C\u591A\u9805\u5F0F\u74B0\u53EF\u4EE5\u88AB\u523B\u5283\u70BA\u4EA4\u63DB -\u4EE3\u6578\u7BC4\u7587\u4E2D\u7684\u81EA\u7531\u5C0D\u8C61\u3002"@zh . "Wenn ein kommutativer Ring mit einer ist, dann ist der Polynomring die Menge aller Polynome mit Koeffizienten aus dem Ring und der Variablen zusammen mit der \u00FCblichen Addition und Multiplikation von Polynomen. Davon zu unterscheiden sind in der abstrakten Algebra die Polynomfunktionen, nicht zuletzt, weil unterschiedliche Polynome dieselbe Polynomfunktion induzieren k\u00F6nnen."@de . . . . . . . . . . "52203"^^ . . "En alg\u00E8bre, le terme de polyn\u00F4me formel, ou simplement polyn\u00F4me, est le nom g\u00E9n\u00E9rique donn\u00E9 aux \u00E9l\u00E9ments d'une structure construite \u00E0 partir d'un ensemble de nombres. On consid\u00E8re un ensemble A de nombres, qui peut \u00EAtre celui des entiers ou des r\u00E9els, et on lui adjoint un \u00E9l\u00E9ment X, appel\u00E9 ind\u00E9termin\u00E9e. La structure est constitu\u00E9e par les nombres, le polyn\u00F4me X, les puissances de X multipli\u00E9es par un nombre, aussi appel\u00E9s mon\u00F4mes (de la forme aXn), ainsi que les sommes de mon\u00F4mes. La structure est g\u00E9n\u00E9ralement not\u00E9e A[X]. Les r\u00E8gles de notation de l'addition et de la multiplication ne sont pas modifi\u00E9es dans la nouvelle structure, ainsi X + X est not\u00E9 2.X, ou encore X.X est not\u00E9 X2. Des exemples de polyn\u00F4mes formels sont : L'ensemble A, utilis\u00E9 pour b\u00E2tir la structure A[X], peut \u00EAtre compos\u00E9 de nombres, mais ce n'est pas indispensable. On lui demande seulement de supporter deux op\u00E9rations : l'addition et la multiplication. Si ces deux op\u00E9rations poss\u00E8dent certaines propri\u00E9t\u00E9s comme l'associativit\u00E9, la commutativit\u00E9 et la distributivit\u00E9 de la multiplication sur l'addition, on dit que A est un anneau commutatif. On lui demande souvent de poss\u00E9der un \u00E9l\u00E9ment neutre pour la multiplication. Seul ce cas est trait\u00E9 dans cet article. Parfois, A poss\u00E8de des propri\u00E9t\u00E9s encore plus fortes, comme d'\u00EAtre un corps commutatif, ce qui signifie que tout \u00E9l\u00E9ment diff\u00E9rent de 0 est inversible pour la multiplication, \u00E0 l'image des rationnels ou des r\u00E9els. Dans ce cas, en plus de l'addition et de la multiplication, la structure A[X] poss\u00E8de une division euclidienne, \u00E0 l'image de l'anneau des entiers et il devient possible d'utiliser les techniques de l'arithm\u00E9tique \u00E9l\u00E9mentaire pour travailler sur les polyn\u00F4mes formels. L'identit\u00E9 de B\u00E9zout s'applique, comme le lemme d'Euclide ou le th\u00E9or\u00E8me fondamental de l'arithm\u00E9tique. Il existe un \u00E9quivalent des nombres premiers constitu\u00E9 par les polyn\u00F4mes unitaires irr\u00E9ductibles. Quelle que soit la nature de l'anneau commutatif et unitaire A, la structure A[X] poss\u00E8de au moins les caract\u00E9ristiques d'un anneau commutatif. On parle d'anneau des polyn\u00F4mes formels. Le polyn\u00F4me formel est un des outils \u00E0 la base de l'alg\u00E8bre. Initialement, il \u00E9tait utilis\u00E9 pour r\u00E9soudre des \u00E9quations dites alg\u00E9briques. R\u00E9soudre l'\u00E9quation alg\u00E9brique revient \u00E0 r\u00E9pondre \u00E0 la question : par quelle valeur doit-on remplacer X pour que l'expression obtenue soit \u00E9gale \u00E0 0 ? Une solution est appel\u00E9e racine du polyn\u00F4me. Le polyn\u00F4me formel est maintenant utilis\u00E9 dans de vastes th\u00E9ories comme la th\u00E9orie de Galois ou la g\u00E9om\u00E9trie alg\u00E9brique et qui d\u00E9passent le cadre de la th\u00E9orie des \u00E9quations. De m\u00EAme que l'anneau A peut \u00EAtre \u00E9tendu \u00E0 une structure plus vaste A[X], l'anneau des polyn\u00F4mes \u00E0 une ind\u00E9termin\u00E9e peut encore \u00EAtre \u00E9tendu, soit par un anneau \u00E0 plusieurs ind\u00E9termin\u00E9es, soit par le corps des fractions rationnelles, soit par l'anneau des s\u00E9ries formelles. Dans toute la suite de l'article, A d\u00E9signe un anneau int\u00E8gre, K un corps commutatif, \u2124 l'anneau des nombres entiers, \u211D le corps des nombres r\u00E9els et \u2102 celui des nombres complexes."@fr . "Wenn ein kommutativer Ring mit einer ist, dann ist der Polynomring die Menge aller Polynome mit Koeffizienten aus dem Ring und der Variablen zusammen mit der \u00FCblichen Addition und Multiplikation von Polynomen. Davon zu unterscheiden sind in der abstrakten Algebra die Polynomfunktionen, nicht zuletzt, weil unterschiedliche Polynome dieselbe Polynomfunktion induzieren k\u00F6nnen."@de . . . . . . . . . . . "O anel de polin\u00F4mios com coeficientes em um anel qualquer e qualquer n\u00FAmero de indeterminadas \u00E9 a generaliza\u00E7\u00E3o dos an\u00E9is como , dos polin\u00F4mios com coeficientes reais p(x) = a0 + a1 x + ... + an xn. De forma gen\u00E9rica, para definir-se o anel dos polin\u00F4mios precisa-se: \n* um anel A dos coeficientes; \n* um conjunto S das indeterminadas. As indeterminadas aqui tem um significado puramente abstrato, n\u00E3o sendo exigido que S tenha nenhuma estrutura. Assim, \u00E9 conveniente que S seja um conjunto de s\u00EDmbolos, e (para evitar ambiguidades) que seja disjunto de A. Um polin\u00F4mio com coeficientes em A e indeterminadas em S pode ser: \n* o polin\u00F4mio nulo, denominado 0 (exceto quando haja necessidade de fazer alguma diferen\u00E7a entre este polin\u00F4mio e o elemento neutro de A; neste caso, podem-se usar \u00EDndices para marcar a diferen\u00E7a entre eles: 0A e 0A[S]). \n* os mon\u00F4mios, que s\u00E3o representados pela justaposi\u00E7\u00E3o de um elemento (n\u00E3o-nulo) de A seguido de um n\u00FAmero finito de elementos de S (podendo ser nenhum) elevados a uma pot\u00EAncia inteira positiva. Por exemplo, se e S = {x, y}, ent\u00E3o 2, 2 x1 e 2 x\u00B2 y\u00B3 s\u00E3o mon\u00F4mios. Aqui \u00E9 importante notar que os produtos de pot\u00EAncias de S comutam, por exemplo, 2 x\u00B2 y\u00B3 = 2 y\u00B3 x\u00B2. Quando a pot\u00EAncia for um, representa-se o mon\u00F4mio sem este valor: 2 x\u00B2 y1 = 2 x\u00B2 y. \n* uma soma de dois ou mais mon\u00F4mios (mas sempre uma quantidade finita), em que a parte indeterminada de todas parcelas s\u00E3o diferentes. Novamente, esta soma \u00E9 comutativa, de forma que duas somas que diferem por uma permuta\u00E7\u00E3o das parcelas s\u00E3o iguais. O anel de polin\u00F4mios \u00E9 este conjunto A[S] com duas opera\u00E7\u00F5es de soma de polin\u00F4mios e produto de polin\u00F4mios, definidas de forma que: \n* o polin\u00F4mio nulo \u00E9 elemento neutro aditivo \n* A[S] \u00E9 um anel \n* o produto de mon\u00F4mios se comporta como se as indeterminadas comutassem entre si, e que o produto de xn e xm seja xn + m Existem v\u00E1rias formas equivalentes de criar modelos para A[S], por exemplo o conjunto de todos os objetos ,[1] onde , , cada -tupla de \u00E9 diferente para diferente valor de , pode servir de modelo para o anel de polin\u00F4mios com indeterminadas em sobre . \u00C9 importante notar que essa express\u00E3o \u00E9 puramente formal, n\u00E3o significando nenhuma opera\u00E7\u00E3o interna dos elementos de S. No caso particular em que m = 0, temos o polin\u00F4mio nulo, tamb\u00E9m representado por 0. No caso particular m = 1, temos um mon\u00F4mio. No caso particular m = 1 e n = 0, temos um elemento de A sendo usado para representar um elemento de A[S]."@pt . . . "Anel de polin\u00F4mios"@pt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "Pier\u015Bcie\u0144 wielomian\u00F3w"@pl . . . "\u062D\u0644\u0642\u0629 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F"@ar . "\uB2E4\uD56D\uC2DD\uD658"@ko . . . . "Polynomring"@sv . "\u6570\u5B66\u3001\u6B8A\u306B\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u591A\u9805\u5F0F\u74B0\uFF08\u305F\u3053\u3046\u3057\u304D\u304B\u3093\u3001\u82F1\u8A9E: polynomial ring\uFF09\u306F\u74B0\u306B\u4FC2\u6570\u3092\u6301\u3064\u4E00\u5909\u6570\u307E\u305F\u306F\u591A\u5909\u6570\u306E\u591A\u9805\u5F0F\u306E\u5168\u4F53\u306E\u96C6\u5408\u304C\u6210\u3059\u74B0\u3067\u3042\u308B\u3002\u591A\u9805\u5F0F\u74B0\u306F\u30D2\u30EB\u30D9\u30EB\u30C8\u306E\u57FA\u5E95\u5B9A\u7406\u3084\u5206\u89E3\u4F53\u306E\u69CB\u6210\u3001\u7DDA\u578B\u4F5C\u7528\u7D20\u306E\u7406\u89E3\u306A\u3069\u6570\u5B66\u306E\u304B\u306A\u308A\u5E83\u3044\u5206\u91CE\u306B\u5F71\u97FF\u3092\u3082\u3064\u6982\u5FF5\u3067\u3042\u308B\u3002\u306E\u3088\u3046\u306A\u591A\u304F\u306E\u91CD\u8981\u306A\u4E88\u60F3\u304C\u3001\u4ED6\u306E\u74B0\u306E\u7814\u7A76\u306B\u5F71\u97FF\u3092\u3082\u3061\u7FA4\u74B0\u3084\u5F62\u5F0F\u51AA\u7D1A\u6570\u74B0\u306E\u3088\u3046\u306A\u307B\u304B\u306E\u74B0\u306E\u5B9A\u7FA9\u306B\u3055\u3048\u5F71\u97FF\u3092\u53CA\u307C\u3057\u3066\u3044\u308B\u3002"@ja . . . . . . . . . "En matem\u00E0tiques, especialment en el camp de l'\u00E0lgebra abstracta, un anell de polinomis o \u00E0lgebra de polinomis \u00E9s un anell (que tamb\u00E9 \u00E9s una \u00E0lgebra commutativa) format a partir del conjunt de polinomis en una o m\u00E9s variables (o ) amb coeficients en un altre anell, sovint un cos. Els anells de polinomis s'han tractat bastament en diversos \u00E0mbits de les matem\u00E0tiques, com per exemple al , a la construcci\u00F3 de cossos de descomposici\u00F3, o a la comprensi\u00F3 del funcionament dels operadors lineals. Moltes conjectures importants tenen a veure amb anells de polinomis, com la , i han influ\u00EFt l'estudi d'altres tipus d'anells, com els o els anells de s\u00E8ries formals de pot\u00E8ncies."@ca . . . . . . . . "Anillo de polinomios"@es . . . . . . . . "In de ringtheorie, een deelgebied van de wiskunde, is een veeltermring een verzameling van veeltermen in een of meer veranderlijken met co\u00EBffici\u00EBnten in een ring."@nl . "En polynomring \u00E4r inom matematik en ring konstruerad fr\u00E5n en annan ring som kan ses som m\u00E4ngden av alla polynom i ett fixt antal variabler med koefficienter i den ursprungliga ringen."@sv . . . . . . . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0648\u0628\u0627\u0644\u062A\u062D\u062F\u064A\u062F \u0641\u064A \u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062A\u062C\u0631\u064A\u062F\u064A\u060C \u062D\u0644\u0642\u0629 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0647\u064A \u062D\u0644\u0642\u0629 \u0645\u0643\u0648\u0646\u0629 \u0645\u0646 \u0645\u062C\u0645\u0648\u0639\u0629 \u0645\u0646 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0628\u0645\u062A\u063A\u064A\u0631 \u0648\u0627\u062D\u062F \u0623\u0648 \u0628\u0639\u062F\u0629 \u0645\u062A\u063A\u064A\u0631\u0627\u062A\u060C \u0628\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u062A\u0646\u062A\u0645\u064A \u0625\u0644\u0649 \u062D\u0644\u0642\u0629 \u0623\u062E\u0631\u0649\u060C \u0639\u0627\u062F\u0629 \u0645\u0627 \u062A\u0643\u0648\u0646 \u062D\u0642\u0644\u0627. \u0639\u0627\u062F\u0629 \u0645\u0627 \u064A\u0634\u064A\u0631 \u0645\u0635\u0637\u0644\u062D \u062D\u0644\u0642\u0629 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0625\u0644\u0649 \u0627\u0644\u062D\u0627\u0644\u0629 \u0627\u0644\u062E\u0627\u0635\u0629 \u062D\u064A\u062B \u062A\u0643\u0648\u0646 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0647\u0630\u0647 \u0623\u062D\u0627\u062F\u064A\u0629 \u0627\u0644\u0645\u062A\u063A\u064A\u0631\u060C \u0648\u062D\u064A\u062B \u062A\u0643\u0648\u0646 \u0627\u0644\u062D\u0644\u0642\u0629 \u0627\u0644\u062A\u064A \u062A\u0646\u062A\u0645\u064A \u0625\u0644\u064A\u0647\u0627 \u0627\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u062D\u0642\u0644\u0627. \u062A\u0646\u0628\u062B\u0642 \u0623\u0647\u0645\u064A\u0629 \u0647\u0630\u0647 \u0627\u0644\u062D\u0644\u0642\u0627\u062A \u0645\u0646 \u0627\u0644\u0639\u062F\u062F \u0627\u0644\u0643\u0628\u064A\u0631 \u0645\u0646 \u0627\u0644\u062E\u0635\u0627\u0626\u0635 \u0627\u0644\u062A\u064A \u062A\u0634\u062A\u0631\u0643 \u0641\u064A\u0647\u0627 \u0645\u0639 \u062D\u0644\u0642\u0629 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u0635\u062D\u064A\u062D\u0629. \u062A\u0638\u0647\u0631 \u062D\u0644\u0642\u0627\u062A \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F\u060C \u0645\u0634\u064E\u0643\u0644\u0629\u064B \u062C\u0632\u0621\u0627 \u0645\u064F\u0647\u0645\u0627\u060C \u0641\u064A \u0627\u0644\u0639\u062F\u064A\u062F \u0645\u0646 \u0645\u062C\u0627\u0644\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A. \u0646\u0638\u0631\u064A\u0629 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0648\u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062A\u0628\u0627\u062F\u0644\u064A \u0648\u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u062C\u0628\u0631\u064A\u0629 \u0623\u0645\u062B\u0644\u0629 \u0639\u0644\u0649 \u0630\u0644\u0643."@ar . . . "In algebra astratta, l'anello dei polinomi costruiti a partire da un certo anello \u00E8 una struttura algebrica contenente tutte le espressioni polinomiali a coefficienti in . Se \u00E8 un dominio d'integrit\u00E0, il suo campo dei quozienti \u00E8 dato dall'insieme delle funzioni razionali a coefficienti nel campo dei quozienti di ."@it . . . "Polynomial ring"@en . . "\u041A\u043E\u043B\u044C\u0446\u043E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u043E\u0432"@ru . . . . . . . . . . . . . . "In de ringtheorie, een deelgebied van de wiskunde, is een veeltermring een verzameling van veeltermen in een of meer veranderlijken met co\u00EBffici\u00EBnten in een ring."@nl . . . "Polynomi\u00E1ln\u00ED okruh"@cs . . . . . . "1099966446"^^ . . . . . . . . . . "\u041A\u0456\u043B\u044C\u0446\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432 \u2014 \u043A\u0456\u043B\u044C\u0446\u0435 \u0432 \u0430\u0431\u0441\u0442\u0440\u0430\u043A\u0442\u043D\u0456\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0456, \u0443\u0442\u0432\u043E\u0440\u0435\u043D\u0435 \u043C\u043D\u043E\u0436\u0438\u043D\u043E\u044E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432 (\u043E\u0434\u043D\u0456\u0454\u0457 \u0430\u0431\u043E \u0434\u0435\u043A\u0456\u043B\u044C\u043A\u043E\u0445 \u0437\u043C\u0456\u043D\u043D\u0438\u0445) \u0437 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0430\u043C\u0438 \u0437 \u0434\u0435\u044F\u043A\u043E\u0433\u043E \u0456\u043D\u0448\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F. \u041A\u0456\u043B\u044C\u0446\u044F \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432 \u0432\u0456\u0434\u0456\u0433\u0440\u0430\u044E\u0442\u044C \u0432\u0430\u0436\u043B\u0438\u0432\u0443 \u0440\u043E\u043B\u044C \u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0446\u0456, \u0432\u0456\u0434 \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u0413\u0456\u043B\u044C\u0431\u0435\u0440\u0442\u0430 \u043F\u0440\u043E \u0431\u0430\u0437\u0438\u0441 \u0456 \u043F\u043E\u0431\u0443\u0434\u043E\u0432\u0438 \u043F\u043E\u043B\u0456\u0432 \u0440\u043E\u0437\u043A\u043B\u0430\u0434\u0443 \u0434\u043E \u0440\u043E\u0437\u0443\u043C\u0456\u043D\u043D\u044F \u043B\u0456\u043D\u0456\u0439\u043D\u043E\u0433\u043E \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0430."@uk . . . . . . . . . . . . . "Veeltermring"@nl . . . "Pier\u015Bcie\u0144 wielomian\u00F3w \u2013 pier\u015Bcie\u0144 okre\u015Blony na zbiorze wielomian\u00F3w jednej lub wi\u0119cej zmiennych o wsp\u00F3\u0142czynnikach z ustalonego pier\u015Bcienia. Pier\u015Bcienie wielomian\u00F3w stanowi\u0142y inspiracj\u0119 do rozwoju wielu dzia\u0142\u00F3w matematyki, pocz\u0105wszy od twierdzenia Hilberta o bazie, przez konstrukcj\u0119 , po rozumienie operatora liniowego. Wiele wa\u017Cnych hipotez, takich jak , wp\u0142yn\u0119\u0142o na badania nad innymi rodzajami pier\u015Bcieni, a nawet by\u0142o \u017Ar\u00F3d\u0142em nowych definicji pier\u015Bcieni, takich jak , czy ."@pl . . . "\u5728\u62BD\u8C61\u4EE3\u6578\u4E2D\uFF0C\u591A\u9805\u5F0F\u74B0\u63A8\u5EE3\u4E86\u521D\u7B49\u6578\u5B78\u4E2D\u7684\u591A\u9805\u5F0F\u3002\u4E00\u500B\u74B0 \u4E0A\u7684\u591A\u9805\u5F0F\u74B0\u662F\u7531\u4FC2\u6578\u5728 \u4E2D\u7684\u591A\u9805\u5F0F\u69CB\u6210\u7684\u74B0\uFF0C\u5176\u4E2D\u7684\u4EE3\u6578\u904B\u7B97\u7531\u591A\u9805\u5F0F\u7684\u4E58\u6CD5\u8207\u52A0\u6CD5\u5B9A\u7FA9\u3002\u5728\u7BC4\u7587\u8AD6\u7684\u8A9E\u8A00\u4E2D\uFF0C\u7576 \u70BA\u4EA4\u63DB\u74B0\u6642\uFF0C\u591A\u9805\u5F0F\u74B0\u53EF\u4EE5\u88AB\u523B\u5283\u70BA\u4EA4\u63DB -\u4EE3\u6578\u7BC4\u7587\u4E2D\u7684\u81EA\u7531\u5C0D\u8C61\u3002"@zh . . . . . . . . . . "\u041A\u0456\u043B\u044C\u0446\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432 \u2014 \u043A\u0456\u043B\u044C\u0446\u0435 \u0432 \u0430\u0431\u0441\u0442\u0440\u0430\u043A\u0442\u043D\u0456\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0456, \u0443\u0442\u0432\u043E\u0440\u0435\u043D\u0435 \u043C\u043D\u043E\u0436\u0438\u043D\u043E\u044E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432 (\u043E\u0434\u043D\u0456\u0454\u0457 \u0430\u0431\u043E \u0434\u0435\u043A\u0456\u043B\u044C\u043A\u043E\u0445 \u0437\u043C\u0456\u043D\u043D\u0438\u0445) \u0437 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0430\u043C\u0438 \u0437 \u0434\u0435\u044F\u043A\u043E\u0433\u043E \u0456\u043D\u0448\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F. \u041A\u0456\u043B\u044C\u0446\u044F \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432 \u0432\u0456\u0434\u0456\u0433\u0440\u0430\u044E\u0442\u044C \u0432\u0430\u0436\u043B\u0438\u0432\u0443 \u0440\u043E\u043B\u044C \u0432 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0446\u0456, \u0432\u0456\u0434 \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u0413\u0456\u043B\u044C\u0431\u0435\u0440\u0442\u0430 \u043F\u0440\u043E \u0431\u0430\u0437\u0438\u0441 \u0456 \u043F\u043E\u0431\u0443\u0434\u043E\u0432\u0438 \u043F\u043E\u043B\u0456\u0432 \u0440\u043E\u0437\u043A\u043B\u0430\u0434\u0443 \u0434\u043E \u0440\u043E\u0437\u0443\u043C\u0456\u043D\u043D\u044F \u043B\u0456\u043D\u0456\u0439\u043D\u043E\u0433\u043E \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0430."@uk . . . . . . "\u6570\u5B66\u3001\u6B8A\u306B\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u591A\u9805\u5F0F\u74B0\uFF08\u305F\u3053\u3046\u3057\u304D\u304B\u3093\u3001\u82F1\u8A9E: polynomial ring\uFF09\u306F\u74B0\u306B\u4FC2\u6570\u3092\u6301\u3064\u4E00\u5909\u6570\u307E\u305F\u306F\u591A\u5909\u6570\u306E\u591A\u9805\u5F0F\u306E\u5168\u4F53\u306E\u96C6\u5408\u304C\u6210\u3059\u74B0\u3067\u3042\u308B\u3002\u591A\u9805\u5F0F\u74B0\u306F\u30D2\u30EB\u30D9\u30EB\u30C8\u306E\u57FA\u5E95\u5B9A\u7406\u3084\u5206\u89E3\u4F53\u306E\u69CB\u6210\u3001\u7DDA\u578B\u4F5C\u7528\u7D20\u306E\u7406\u89E3\u306A\u3069\u6570\u5B66\u306E\u304B\u306A\u308A\u5E83\u3044\u5206\u91CE\u306B\u5F71\u97FF\u3092\u3082\u3064\u6982\u5FF5\u3067\u3042\u308B\u3002\u306E\u3088\u3046\u306A\u591A\u304F\u306E\u91CD\u8981\u306A\u4E88\u60F3\u304C\u3001\u4ED6\u306E\u74B0\u306E\u7814\u7A76\u306B\u5F71\u97FF\u3092\u3082\u3061\u7FA4\u74B0\u3084\u5F62\u5F0F\u51AA\u7D1A\u6570\u74B0\u306E\u3088\u3046\u306A\u307B\u304B\u306E\u74B0\u306E\u5B9A\u7FA9\u306B\u3055\u3048\u5F71\u97FF\u3092\u53CA\u307C\u3057\u3066\u3044\u308B\u3002"@ja . . . "\uB300\uC218\uD559\uC5D0\uC11C \uB2E4\uD56D\uC2DD\uD658(\u591A\u9805\u5F0F\u74B0, \uC601\uC5B4: polynomial ring)\uC740 \uC5B4\uB5A4 \uC8FC\uC5B4\uC9C4 \uD658\uC744 \uACC4\uC218\uB85C \uD558\uB294 \uB2E4\uD56D\uC2DD\uB4E4\uB85C \uAD6C\uC131\uB41C \uD658\uC774\uB2E4."@ko . . . . . "Sea un anillo y cualquier conjunto. El conjunto contiene los elementos de la forma:, en donde , , y cada -tupla de n\u00FAmeros naturales es diferente para diferente valor de , se dice anillo de polinomios con indeterminadas en sobre ."@es . . . . "Polyn\u00F4me formel"@fr . . . "\uB300\uC218\uD559\uC5D0\uC11C \uB2E4\uD56D\uC2DD\uD658(\u591A\u9805\u5F0F\u74B0, \uC601\uC5B4: polynomial ring)\uC740 \uC5B4\uB5A4 \uC8FC\uC5B4\uC9C4 \uD658\uC744 \uACC4\uC218\uB85C \uD558\uB294 \uB2E4\uD56D\uC2DD\uB4E4\uB85C \uAD6C\uC131\uB41C \uD658\uC774\uB2E4."@ko . . "In algebra astratta, l'anello dei polinomi costruiti a partire da un certo anello \u00E8 una struttura algebrica contenente tutte le espressioni polinomiali a coefficienti in . Se \u00E8 un dominio d'integrit\u00E0, il suo campo dei quozienti \u00E8 dato dall'insieme delle funzioni razionali a coefficienti nel campo dei quozienti di ."@it . . . . . "In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term \"polynomial ring\" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers."@en . . . . . "Polynomi\u00E1ln\u00ED okruh (t\u00E9\u017E okruh mnoho\u010Dlen\u016F) je v matematice, zejm\u00E9na v algeb\u0159e, takov\u00FD okruh, kter\u00FD je tvo\u0159en mno\u017Einou polynom\u016F s koeficienty z n\u011Bjak\u00E9ho jin\u00E9ho okruhu. Jedn\u00E1 se o d\u016Fle\u017Eit\u00FD algebraick\u00FD koncept a lze se s n\u00EDm setkat nap\u0159\u00EDklad p\u0159i konstrukci rozkladov\u00FDch t\u011Bles nebo v Hilbertov\u011B v\u011Bt\u011B o b\u00E1zi. Okruh polynom\u016F v jedn\u00E9 prom\u011Bnn\u00E9 nad okruhem R je zna\u010Den R[x], okruh ve dvou prom\u011Bnn\u00FDch R[x,y] a tak d\u00E1le."@cs . . . "Polynomi\u00E1ln\u00ED okruh (t\u00E9\u017E okruh mnoho\u010Dlen\u016F) je v matematice, zejm\u00E9na v algeb\u0159e, takov\u00FD okruh, kter\u00FD je tvo\u0159en mno\u017Einou polynom\u016F s koeficienty z n\u011Bjak\u00E9ho jin\u00E9ho okruhu. Jedn\u00E1 se o d\u016Fle\u017Eit\u00FD algebraick\u00FD koncept a lze se s n\u00EDm setkat nap\u0159\u00EDklad p\u0159i konstrukci rozkladov\u00FDch t\u011Bles nebo v Hilbertov\u011B v\u011Bt\u011B o b\u00E1zi. Okruh polynom\u016F v jedn\u00E9 prom\u011Bnn\u00E9 nad okruhem R je zna\u010Den R[x], okruh ve dvou prom\u011Bnn\u00FDch R[x,y] a tak d\u00E1le."@cs . . . . "Pier\u015Bcie\u0144 wielomian\u00F3w \u2013 pier\u015Bcie\u0144 okre\u015Blony na zbiorze wielomian\u00F3w jednej lub wi\u0119cej zmiennych o wsp\u00F3\u0142czynnikach z ustalonego pier\u015Bcienia. Pier\u015Bcienie wielomian\u00F3w stanowi\u0142y inspiracj\u0119 do rozwoju wielu dzia\u0142\u00F3w matematyki, pocz\u0105wszy od twierdzenia Hilberta o bazie, przez konstrukcj\u0119 , po rozumienie operatora liniowego. Wiele wa\u017Cnych hipotez, takich jak , wp\u0142yn\u0119\u0142o na badania nad innymi rodzajami pier\u015Bcieni, a nawet by\u0142o \u017Ar\u00F3d\u0142em nowych definicji pier\u015Bcieni, takich jak , czy ."@pl . . . . . "En alg\u00E8bre, le terme de polyn\u00F4me formel, ou simplement polyn\u00F4me, est le nom g\u00E9n\u00E9rique donn\u00E9 aux \u00E9l\u00E9ments d'une structure construite \u00E0 partir d'un ensemble de nombres. On consid\u00E8re un ensemble A de nombres, qui peut \u00EAtre celui des entiers ou des r\u00E9els, et on lui adjoint un \u00E9l\u00E9ment X, appel\u00E9 ind\u00E9termin\u00E9e. La structure est constitu\u00E9e par les nombres, le polyn\u00F4me X, les puissances de X multipli\u00E9es par un nombre, aussi appel\u00E9s mon\u00F4mes (de la forme aXn), ainsi que les sommes de mon\u00F4mes. La structure est g\u00E9n\u00E9ralement not\u00E9e A[X]. Les r\u00E8gles de notation de l'addition et de la multiplication ne sont pas modifi\u00E9es dans la nouvelle structure, ainsi X + X est not\u00E9 2.X, ou encore X.X est not\u00E9 X2. Des exemples de polyn\u00F4mes formels sont :"@fr . . . . "373065"^^ . . . . . "Anell de polinomis"@ca . . "\u041A\u043E\u043B\u044C\u0446\u043E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u043E\u0432 \u2014 \u043A\u043E\u043B\u044C\u0446\u043E, \u043E\u0431\u0440\u0430\u0437\u043E\u0432\u0430\u043D\u043D\u043E\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0430\u043C\u0438 \u043E\u0442 \u043E\u0434\u043D\u043E\u0439 \u0438\u043B\u0438 \u043D\u0435\u0441\u043A\u043E\u043B\u044C\u043A\u0438\u0445 \u043F\u0435\u0440\u0435\u043C\u0435\u043D\u043D\u044B\u0445 \u0441 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u0430\u043C\u0438 \u0438\u0437 \u0434\u0440\u0443\u0433\u043E\u0433\u043E \u043A\u043E\u043B\u044C\u0446\u0430. \u0418\u0437\u0443\u0447\u0435\u043D\u0438\u0435 \u0441\u0432\u043E\u0439\u0441\u0442\u0432 \u043A\u043E\u043B\u0435\u0446 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u043E\u0432 \u043E\u043A\u0430\u0437\u0430\u043B\u043E \u0431\u043E\u043B\u044C\u0448\u043E\u0435 \u0432\u043B\u0438\u044F\u043D\u0438\u0435 \u043D\u0430 \u043C\u043D\u043E\u0433\u0438\u0435 \u043E\u0431\u043B\u0430\u0441\u0442\u0438 \u0441\u043E\u0432\u0440\u0435\u043C\u0435\u043D\u043D\u043E\u0439 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438; \u043C\u043E\u0436\u043D\u043E \u043F\u0440\u0438\u0432\u0435\u0441\u0442\u0438 \u043F\u0440\u0438\u043C\u0435\u0440\u044B \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u0413\u0438\u043B\u044C\u0431\u0435\u0440\u0442\u0430 \u043E \u0431\u0430\u0437\u0438\u0441\u0435, \u043A\u043E\u043D\u0441\u0442\u0440\u0443\u043A\u0446\u0438\u0438 \u043F\u043E\u043B\u044F \u0440\u0430\u0437\u043B\u043E\u0436\u0435\u043D\u0438\u044F \u0438 \u0438\u0437\u0443\u0447\u0435\u043D\u0438\u044F \u0441\u0432\u043E\u0439\u0441\u0442\u0432 \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u0445 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u043E\u0432."@ru . "Anello dei polinomi"@it . . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0648\u0628\u0627\u0644\u062A\u062D\u062F\u064A\u062F \u0641\u064A \u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062A\u062C\u0631\u064A\u062F\u064A\u060C \u062D\u0644\u0642\u0629 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0647\u064A \u062D\u0644\u0642\u0629 \u0645\u0643\u0648\u0646\u0629 \u0645\u0646 \u0645\u062C\u0645\u0648\u0639\u0629 \u0645\u0646 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0628\u0645\u062A\u063A\u064A\u0631 \u0648\u0627\u062D\u062F \u0623\u0648 \u0628\u0639\u062F\u0629 \u0645\u062A\u063A\u064A\u0631\u0627\u062A\u060C \u0628\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u062A\u0646\u062A\u0645\u064A \u0625\u0644\u0649 \u062D\u0644\u0642\u0629 \u0623\u062E\u0631\u0649\u060C \u0639\u0627\u062F\u0629 \u0645\u0627 \u062A\u0643\u0648\u0646 \u062D\u0642\u0644\u0627. \u0639\u0627\u062F\u0629 \u0645\u0627 \u064A\u0634\u064A\u0631 \u0645\u0635\u0637\u0644\u062D \u062D\u0644\u0642\u0629 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0625\u0644\u0649 \u0627\u0644\u062D\u0627\u0644\u0629 \u0627\u0644\u062E\u0627\u0635\u0629 \u062D\u064A\u062B \u062A\u0643\u0648\u0646 \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F \u0647\u0630\u0647 \u0623\u062D\u0627\u062F\u064A\u0629 \u0627\u0644\u0645\u062A\u063A\u064A\u0631\u060C \u0648\u062D\u064A\u062B \u062A\u0643\u0648\u0646 \u0627\u0644\u062D\u0644\u0642\u0629 \u0627\u0644\u062A\u064A \u062A\u0646\u062A\u0645\u064A \u0625\u0644\u064A\u0647\u0627 \u0627\u0644\u0645\u0639\u0627\u0645\u0644\u0627\u062A \u062D\u0642\u0644\u0627. \u062A\u0646\u0628\u062B\u0642 \u0623\u0647\u0645\u064A\u0629 \u0647\u0630\u0647 \u0627\u0644\u062D\u0644\u0642\u0627\u062A \u0645\u0646 \u0627\u0644\u0639\u062F\u062F \u0627\u0644\u0643\u0628\u064A\u0631 \u0645\u0646 \u0627\u0644\u062E\u0635\u0627\u0626\u0635 \u0627\u0644\u062A\u064A \u062A\u0634\u062A\u0631\u0643 \u0641\u064A\u0647\u0627 \u0645\u0639 \u062D\u0644\u0642\u0629 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u0635\u062D\u064A\u062D\u0629. \u062A\u0638\u0647\u0631 \u062D\u0644\u0642\u0627\u062A \u0645\u062A\u0639\u062F\u062F\u0627\u062A \u0627\u0644\u062D\u062F\u0648\u062F\u060C \u0645\u0634\u064E\u0643\u0644\u0629\u064B \u062C\u0632\u0621\u0627 \u0645\u064F\u0647\u0645\u0627\u060C \u0641\u064A \u0627\u0644\u0639\u062F\u064A\u062F \u0645\u0646 \u0645\u062C\u0627\u0644\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A. \u0646\u0638\u0631\u064A\u0629 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0648\u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062A\u0628\u0627\u062F\u0644\u064A \u0648\u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u062C\u0628\u0631\u064A\u0629 \u0623\u0645\u062B\u0644\u0629 \u0639\u0644\u0649 \u0630\u0644\u0643."@ar . . . . . . . . . . . . . . "En polynomring \u00E4r inom matematik en ring konstruerad fr\u00E5n en annan ring som kan ses som m\u00E4ngden av alla polynom i ett fixt antal variabler med koefficienter i den ursprungliga ringen."@sv . "\u591A\u9805\u5F0F\u74B0"@ja . . . . "In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term \"polynomial ring\" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety."@en . . . . . . "\u041A\u043E\u043B\u044C\u0446\u043E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u043E\u0432 \u2014 \u043A\u043E\u043B\u044C\u0446\u043E, \u043E\u0431\u0440\u0430\u0437\u043E\u0432\u0430\u043D\u043D\u043E\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0430\u043C\u0438 \u043E\u0442 \u043E\u0434\u043D\u043E\u0439 \u0438\u043B\u0438 \u043D\u0435\u0441\u043A\u043E\u043B\u044C\u043A\u0438\u0445 \u043F\u0435\u0440\u0435\u043C\u0435\u043D\u043D\u044B\u0445 \u0441 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u0430\u043C\u0438 \u0438\u0437 \u0434\u0440\u0443\u0433\u043E\u0433\u043E \u043A\u043E\u043B\u044C\u0446\u0430. \u0418\u0437\u0443\u0447\u0435\u043D\u0438\u0435 \u0441\u0432\u043E\u0439\u0441\u0442\u0432 \u043A\u043E\u043B\u0435\u0446 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u043E\u0432 \u043E\u043A\u0430\u0437\u0430\u043B\u043E \u0431\u043E\u043B\u044C\u0448\u043E\u0435 \u0432\u043B\u0438\u044F\u043D\u0438\u0435 \u043D\u0430 \u043C\u043D\u043E\u0433\u0438\u0435 \u043E\u0431\u043B\u0430\u0441\u0442\u0438 \u0441\u043E\u0432\u0440\u0435\u043C\u0435\u043D\u043D\u043E\u0439 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438; \u043C\u043E\u0436\u043D\u043E \u043F\u0440\u0438\u0432\u0435\u0441\u0442\u0438 \u043F\u0440\u0438\u043C\u0435\u0440\u044B \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u0413\u0438\u043B\u044C\u0431\u0435\u0440\u0442\u0430 \u043E \u0431\u0430\u0437\u0438\u0441\u0435, \u043A\u043E\u043D\u0441\u0442\u0440\u0443\u043A\u0446\u0438\u0438 \u043F\u043E\u043B\u044F \u0440\u0430\u0437\u043B\u043E\u0436\u0435\u043D\u0438\u044F \u0438 \u0438\u0437\u0443\u0447\u0435\u043D\u0438\u044F \u0441\u0432\u043E\u0439\u0441\u0442\u0432 \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u0445 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u043E\u0432."@ru . . . . . . . . . "\u591A\u9879\u5F0F\u73AF"@zh . "\u041A\u0456\u043B\u044C\u0446\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0456\u0432"@uk . "Sea un anillo y cualquier conjunto. El conjunto contiene los elementos de la forma:, en donde , , y cada -tupla de n\u00FAmeros naturales es diferente para diferente valor de , se dice anillo de polinomios con indeterminadas en sobre ."@es . . . . . . . . . .