arXiv:alg-geom/9305009v1 19 May 1993 ON THE APPROXIMATE ROOTS OF POLYNOMIALS by Janusz Gwo´zdziewicz and Arkadiusz P loski Abstract. We give a simplified approach to the Abhyankar–Moh theory of approxi- mate roots. Our considerations are based on properties of the intersection multiplicity of local curves. 1. The main results For any power series f , g ∈ C  [x,y]  we define the intersection number (f,g) 0 by (f,g) 0 = dim C C  [x,y]  /(f,g). Suppose that f = f (x,y) is an irreducible power se- ries and let n =(f,x) 0 = ord f (0,y) < +∞. Then there exists a power series y(t) ∈ C  [t]  , ord y(t) > 0 such that f (t n ,y(t)) = 0. We have (f,g) 0 = ord g(t n ,y(t)) for any g = g(x,y) ∈ C  [x,y]  . The mapping g → (f,g) 0 induces a valuation v f of the ring C  [x,y]  /(f ). Let Γ(f ) be the semigroup of v f i.e. Γ(f )= { (f,g) 0 ∈ N : g ≡ 0 mod (f ) }. According to the well known structure theorem for the semigroup Γ(f ) ([1], [2], [10] and Sect. 3 of this paper) there is a unique sequence of positive integers ¯ b 0 , ¯ b 1 , ... , ¯ b h such that (i) ¯ b 0 =(f,x) 0 , (ii) ¯ b k = min(Γ(f ) \ (N ¯ b 0 + ··· + N ¯ b k-1 )) for k =1,...,h. (iii) Γ(f )= N ¯ b 0 + ··· + N ¯ b h i.e. Γ(f ) is generated by ¯ b 0 , ¯ b 1 ,..., ¯ b h If the conditions (i), (ii), (iii) are satisfied, we write Γ(f )= 〈 ¯ b 0 ,..., ¯ b h 〉. Also define B k = gcd( ¯ b 0 ,..., ¯ b k ) for k =0, 1,...,h and n k = B k-1 /B k for k =1,...,h. We have (iv) n k > 1 and the sequence B k-1 ¯ b k is strictly increasing for k ≥ 1. Let O be an integral domain of characteristic zero. Let g ∈O[y] be a monic polynomial and let d be a positive integer such that d divides deg g. According to Abhyankar and Moh [2] the approximate d th root of g denoted d √ g is defined to be the unique monic polynomial satisfying deg(g − ( d √ g) d ) < deg g − deg d √ g. Obviously deg d √ g = deg g/d. Let 1 ≤ k ≤ h. Theorem 1.1. Let g = g(x,y) ∈ C  [x]  [y] be a monic polynomial, deg y g = n/B k . Suppose that (f,g) 0 >n k ¯ b k . Then (f, n k √ g) 0 = ¯ b k . The proof of (1.1) is given in Sect. 4 of this paper. We shall follow the methods developed by Abhyankar and Moh in the fundamental paper [2] and simplified by Abhyankar in [1]. Let 1 ≤ k ≤ h + 1. Typeset by A M S-T E X 1