CHAP, xxni] INFINITUDE OF SOLUTIONS OF f(x, y) = 0. 675 



where 0,- is homogeneous and of degree j. (I) Let (f> n (x, y) be arithmetically 

 reducible; let Ci be a simple real root of n (l, c) =0 of degree X; let ^(1, c) 

 be an irreducible factor of </>, of degree k (k<ri) and with the root ci. 

 Then 7^ = has, on the infinite branch whose asymptote has Ci as angular 

 coefficient, an infinitude of solutions only if one of the 0*(1, Ci), i = n l, 

 , n k, is not zero. (II) There exists no irreducible equation F(x, y)=Q 

 with integral coefficients having an infinitude of integral solutions on an 

 infinite branch of F = G such that the angular coefficient of the asymptote 

 is rational and not zero, if this coefficient is a simple root of 0(1, c)=0. 

 If the real angular coefficients of the asymptotes of F = are all rational, not 

 zero and distinct, then F = has only a finite number of integral solutions. 

 By amplifying the case k = 2, he obtains a complicated third theorem ; also 

 one on F(x, y, z) = 0. 



A. Thue 9 proved that, if U(x, y) is an irreducible homogeneous poly- 

 nomial with integral coefficients and c is a given constant, U(p, q)=c has 

 only a finite number of positive integral solutions p, q, when the degree of 

 U exceeds 2. 



A. Thue 10 considered homogeneous integral functions P(x t y}, Q(x, y), 

 R(x, y) of degrees p, q, r, with integral coefficients, P(x, y) being irreducible. 

 If p > q, p > 2, P = Q does not have an infinitude of pairs of integral solutions 

 x, y. If p>q>r, p<q+r, P+Q+R = Q is not satisfied by an infinitude of 

 pairs of relatively prime integers x, y. 



E. Maillet 11 completed a lacuna in the proof by Thue 9 and gave the 

 following generalization of his theorem. Let 0; be a homogeneous poly- 

 nomial of degree i in x, y. While the coefficients of , , <j> 3 need not 

 be rational, let <f>r (r>s) have integral coefficients and contain a term in x r 

 and one in y r . If 



<t>r(x, y) - <f>*(x> y} - 0a-lfo y) 00 = 



is irreducible, it has an infinitude of integral solutions x, y only when s 

 exceeds a specified quantity depending on the reducibility of r = 0. When 

 r is irreducible, this quantity is TI 2 or r\ 1, according as r = 2ri or 

 r = 2ri-fl. 



Maillet lla gave a practical method to find an upper limit to the absolute 

 values of the integral solutions x, y of an equation of type (2), subject to 

 certain conditions on n which imply that (2) has only a finite number of 

 integral solutions. 



RATIONAL POINTS ON THE PLANE CURVE f(x, y, z) = 0. 



D. Hilbert and A. Hurwitz 12 treated homogeneous polynomials f(x i, x z , x 3 ) 

 of degree n with integral coefficients such that the curve /=0 is of genus 

 (or deficiency, geschlecht) zero. In view of results by M. Noether, 13 we 



9 Jour, fur Math., 135, 1909, 303-4. Cf. Maillet. 11 



10 Skrifter Videnskaps. Kristiania (Math.), 1, 1911, No. 3 (German). 



11 Nouv. Ann. Math., (4), 16, 1916, 338-345. 

 lla Ibid., (4), 18, 1918, 281-92. 



12 Acta Math., 14, 1890-1, 217-24. 



13 Math. Annalen, 23, 1884, 311-358. 



