NEWTON POLYHEDRA AND ESTIMATION TO EXPONENTIAL SUMS

Kamel Arifin Mohd Atan

ICMSA
Vol 2, No 1 (2006): Pure Maths : ICMSA 2006

The classical Newton polygon is a device for computing the fractional power series expansions of algebraic functions. Newton gave a number of examples of this process in his ”Method of Fluxions” which amount to a general method. However, it was not till much later that Puiseux proved that every branch of a plane algebraic curve defined by a polynomial equation f(x, y) = 0 has an expansionin a neighbourhood of a point (x0, y0) on the curve. In practice, the integers a, b and q can be read off from the Newton polygon and the coefficients cj can be determined successively with ever-increasing labour.