4 2. Non-singularity and Resolution of Singularities Set r0 = mult(/(0, j/)) mult(/). Set 5o = min : : j TQ and an ^ 0 . [r0-j J £0 = oo if and only if / = uyr°, where u is a unit in K[[x, y]]. Suppose that So oo. Then we can write 2+5o:/£on) with aor0 7^ 0 a n d the weighted leading form Ls0(x,y) = 2^ a ijx%y^ — a 0r0yr° + terms of lower degree in y i+Soj=Soro has at least two non-zero terms. We can thus choose 0 ^ c\ G K so that Ztf0(l,ci)= ^ a y c j = 0 . Write ^0 — ^5 where qo,Po are relatively prime positive integers. We make a transformation x = xf, y = xp1°(y1 + ci). Then / ^ ? P ° / i ( x i , l / i ) , where (2.1) /i(^i,yi) = X ] a ^'( C l + yi)3 +xiH(xiiVi)- i-\-Soj=Soro By our choice of ci, /i(0,0) = 0. Set ri = mult(/i(0, yi)). We see that r\ TQ. We have an expansion h = ^2aij(1)x\yv Set Si — min : : j r\ and a^-(l) 7^ 0 , and write 8\ = ^ with pi,gi relatively prime. We can then choose C 2 Gif for /1 , in the same way that we chose c\ for / , and iterate this process, obtaining a sequence of transformations x = xf, y = x f (yi-hci), (2.2) x i = xl1* 2/i = ^(jfe + c2), Either this sequence of transformations terminates after a finite number n of steps with 5n = 00, or we can construct an infinite sequence of trans- formations with Sn 00 for all n. This allows us to write y as a series in ascending fractional powers of x.

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