Newton polygon [SEP] named after
" by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition.
History.
Newton polygons are named after Isaac Newton, who first described them and some of their uses in correspondence from the year 1676 addressed to Henry Oldenburg.
Applications.
A Newton Polygon is sometimes a special case of a , and can be used to construct asymptotic solutions of two-variable polynomial equations like 
formula_8
Another application of the Newton polygon comes from the following result:
same slope.
The Newton polygon of an "F"-isocrystal.
The Newton polygon of an "F"-isocrystal encodes the dimensions of the pieces of given slope. If the "F"-isocrystal is a sum of isoclinic pieces with slopes "s"  "s"  ... and dimensions (as Witt ring modules) "d", "d"... then the Newton polygon has vertices (0,0), ("x", "y"), ("x", "y")... where the "