Secrion III., 1911. [S1] MRANSAR SAC: 
Algebraic proof of the existence of the power-series representing the 
branches of an algebraic function. 
By CNFIELDS Pai: 
(Read May 17, 1911). 
IN his book* on the algebraic functions the writer has started out 
on assuming the existence of the power-series representing the 
branches of an algebraic function. For the existence of the series in 
question mathematicians have given proofs, both algebraic** and 
transcendental. It has seemed to the writer however that it might 
perhaps add to the completeness of his theory of the algebraic 
functions to here give the algebraic proof of the theorem in question 
in the special form in which he has presented it in his lectures on the 
algebraic functions at the University of Toronto. 
It will suffice to prove the existence of the power-series repre- 
senting the branches of an arbitrarily chosen algebraic function in 
the neighborhood of the value z=o, since any other case reduces 
itself to this case by one or other of the transformations zg—a=s or 
z=s 1 It will also be sufficient to confine our attention to the case 
of an integral algebraic function, since from a non-integral algebraic 
function “ we can always obtain an integral algebraic function on 
multiplying by a polynominal in z. 
We write our integral algebraic equation in the form 
n 
L: ie Cerne) = ÿ f= (3) v= Sz: f. (2) vs = 
s=0o 
where the functions f, (z) are polynomials in z in which the constant 
terms are different from o—apart from the case in which the co- 
efficient of the power of v in question is identically 0. We call c, the 
order of the coefficient of 1 ; in the exceptional case just referred to 
we take c; ==. We evidently have c,=o and we may assume f, (z) =1. 
We shall also assume that the equation (1.) in v presents no repeated 
factors. It may however be reducible or irreducible. We might 
note that our argument in what follows holds also where the functions 
fs (z) are power-series in z. 
In (1.) we now substitute 
2. V=Wy+0; =a2° +7, 
where 
Co — Gy Co — Co Co —C Co—C 
3. c=Max.{ a 2 em} 
I 2 5 n 
and where a is as yet undetermined. It is evident that c is not 
Co—Cn 
negative since c,, and therefore is not negative. For v, we ob- 
tain the equation 
* Theory of the algebraic functions of a complex variable. Berlin, Mayer & Müller, 1906. 
** See Hensel und Landsberg. Theorie der algebraischen Funktionen ei Variabel 73) 
a a ne £ unktionen einer Variabeln (pp. 25-73) 
