In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields.
In the original case, the local field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring ,
over , where 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 terms
of the power series expansion solutions to equations
where is a polynomial with coefficients in , the polynomial ring; that is, implicitly defined algebraic functions. The exponents here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in
with for a denominator corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating .
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.
In mathematics, the Newtonpolygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields...
the Newtonpolygon. Let consider, in a Cartesian plane, the points of coordinates ( i , v ( a i ) ) . {\displaystyle (i,v(a_{i})).} The Newtonpolygon of...
Sir Isaac Newton FRS (25 December 1642 – 20 March 1726/27) was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian...
hull of points in the plane appears, in the form of a Newtonpolygon, in a letter from Isaac Newton to Henry Oldenburg in 1676. The term "convex hull" itself...
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solution. An early example of the dominant balance method is the Newtonpolygon method. Newton developed this method to find an explicit approximation for...
d(L) contains a non-zero lattice point Mixed volume Mixture density Newtonpolygon - a tool for understanding the behaviour of polynomials over local fields...
Poincaré rank (Arscott (1995)). The regularity condition is a kind of Newtonpolygon condition, in the sense that the allowed poles are in a region, when...
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variety in nonzero characteristic such that for all n the slopes of the Newtonpolygon of the nth crystalline cohomology are all n/2 (de Jong 2014). For special...
environment and acted upon by gravity in the vertical direction, its support polygon is a horizontal region over which the center of mass must lie to achieve...
on the Newton line. A regular polygon has an inscribed circle which is tangent to each side of the polygon at its midpoint. In a regular polygon with an...
QWERTY keyboard. With "Shapes", Newton can recognize that the user was attempting to draw a circle, a line, a polygon, etc., and it cleans them up into...
derivatives requires implicit differentiation. Newton's diagram (also known as Newton's parallelogram, after Isaac Newton) is a technique for determining the shape...
personal digital assistant devices developed by Apple Computer for the Newton platform in 1993. Some electronic engineering and the manufacture of Apple's...
with respect to any other two points. (Baker 1922a, vol 1, p. 133) Newtonpolygon The convex hull of the points with coordinates given by the exponents...
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