By A. Campillo

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**Additional resources for Algebroid Curves in Positive Characteristics**

**Sample text**

Representation expansion The of we some have system curve U(z Proof: and Conversely, if U(z the quotient is only if for its parametric Hamburger-Noether the c o n d i t i o n s k((u)) ~ k((Zr) ) , and ) = 1 then k((u)) = k((z r field actually and a primitive on is p r i m i t i v e , imply -- if is we r If the r e p r e s e n t a t y(u) E: k((Zr] ] {x,y} hypothesis which first = I . -- tion two from = y'l ) = L(D) - I = N-t, by the induction 1 (D1) = ( D ' 1) a n d c o n s e q u e n t l y (D)= (O') .

Twisted have ~n-1 equation or (()) X curves. Puiseux transformations. sal quadratic parameter transform rT] 2 +I + t p+I relative to a transversal parameter Let us consider a parametric representation type n × = t 1 r X. _iz < , ideal. - (1) in ape expansion maximal +p+2 to but equations 2 p t r ii / + a ril " " " and t i2 + .... 24 ri2 a i r . ij 0, I ~

2 . 1 . 1 . 1 curve that (x) 1 . x 1 in S complete. ), Proof: and is = p > 0. Let (()) E; k t t' (see representation curve over in a basis k((x)): 51 n-1 n Y + A (X) n-I Y + ... (X) ~ E: k I 0-~ then the degree if integers and one the the are curve [] under given a The i(x) are index be in made which the polynomial. 1 not equation curve A Remark (x) s inseparability the knows whether n i for we , , separablity Consequently, determines not. twisted have ~n-1 equation or (()) X curves. Puiseux transformations.