"Newton polygon"@en . . "In der Mathematik ist das Newton-Polygon ein Werkzeug zur Untersuchung von Polynomen."@de . "28"^^ . . . . . . "In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ringK[[X]], over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading termsaXrof the power series expansion solutions to equationsP(F(X)) = 0where P is a polynomial with coefficients in K[X], the polynomial ring; that is, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series inK[[Y]]with Y = X1/d for a denominator d corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating d.After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves."@en . "En math\u00E9matiques, le polygone de Newton est un polygone du plan euclidien que l'on peut associer \u00E0 un polyn\u00F4me, lorsque les coefficients de ce dernier sont \u00E9l\u00E9ments d'un corps valu\u00E9."@fr . . . . . "In der Mathematik ist das Newton-Polygon ein Werkzeug zur Untersuchung von Polynomen."@de . . "99599396"^^ . . "In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ringK[[X]], over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions."@en . "Polygone de Newton"@fr . . . . . "Newton-Polygon"@de . . . . . . . . . "En math\u00E9matiques, le polygone de Newton est un polygone du plan euclidien que l'on peut associer \u00E0 un polyn\u00F4me, lorsque les coefficients de ce dernier sont \u00E9l\u00E9ments d'un corps valu\u00E9. Le polygone de Newton encode un certain nombre d'informations \u00E0 propos de la factorisation d'un polyn\u00F4me, et la localisation de ses racines.Il est particuli\u00E8rement utile lorsque les coefficients du polyn\u00F4me sont \u00E9l\u00E9ments d'un corps local non-archim\u00E9dien, comme par exemple le corps des nombres p-adiques, ou celui des s\u00E9ries de Laurent sur un corps fini, mais il peut \u00E9galement \u00EAtre utilis\u00E9 avec profit dans l'\u00E9tude des polyn\u00F4mes \u00E0 coefficients rationnels, ou des polyn\u00F4mes en plusieurs ind\u00E9termin\u00E9es."@fr . . . . . . "9745"^^ . . . . . . "6744246"^^ . . . . . .