[MathSciNet entry describing Runge's theorem on the _constructive_ finitude of 
the solution set of certain Diophantine equations in two variables.]
   
   94i:11025 11D41
   Grytczuk, A.(PL-PUZG-IM); Schinzel, A.(PL-PAN)
   On Runge's theorem about Diophantine equations. Sets, graphs and
   numbers (Budapest, 1991), 329--356,
   Colloq. Math. Soc. János Bolyai, 60,
   North-Holland, Amsterdam, 1992.
   
   C. Runge established an early and general result in Diophantine
   equations [J. Reine Angew. Math. 100 (1887), 425--435; Jbuch 19, 76].
   Once "Runge's condition", which pertains to a class of Diophantine
   equations $F(x,y)=0$, is defined, Runge's theorem then takes the
   following form: If Runge's condition holds, then the equation
   $F(x,y)=0$ has only finitely many integer solutions. Moreover, Runge's
   method of proof furnishes a constructive method for solving the
   Diophantine equation. Various formulations of Runge's condition
   appear, namely, in the 1887 paper of Runge, a paper of the reviewer
   and E. G. Straus [Trans. Amer. Math. Soc. 280 (1983), no. 2, 637--657;
   MR 85c:11031], the paper under review and a book by L. J. Mordell
   [Diophantine equations, Academic Press, London, 1969; MR 40 #2600]. In
   the paper by the reviewer and Straus, Runge's condition is defined as
   follows. Let $$F(x,y)=\sum\sp {d\sb 1}\sb {i=0}\sum\sp {d\sb 2}\sb
   {j=0}a\sb {ij}x\sp iy\sp j$$ be a polynomial in $x$ and $y$, of degree
   $d\sb 1$ and $d\sb 2$ in $x$ and $y$, respectively. Let $\lambda$ be
   any positive real number. The $\lambda$-leading part of $F(x,y)$,
   denoted by $F\sb \lambda(x,y)$, is defined to be the polynomial
   consisting of the sum of all nonzero terms $a\sb {ij}x\sp iy\sp j$ of
   $F(x,y)$ for which $i+\lambda j$ is maximal, for that fixed value of
   $\lambda$. The leading part of $F(x,y)$, denoted by $\widetilde
   F(x,y)$, is defined to be the polynomial consisting of the sum of all
   such terms, as $\lambda$ varies. Note that a related notion is that of
   the leading form of $F(x,y)$, which is the polynomial consisting of
   the sum of all terms of $F(x,y)$ of maximal degree. It is, in this
   notation, $F\sb 1(x,y)$. If the polynomial $F(x,y)$ with integer
   coefficients is irreducible, we say, following the paper of the
   reviewer and Straus, that it satisfies Runge's condition unless, for
   some $\lambda\sb 0$, $\widetilde F(x,y)=F\sb {\lambda\sb 0}(x,y)$ is a
   constant multiple of a power of an irreducible polynomial. That is,
   the polynomial $F(x,y)$ with integer coefficients satisfies Runge's
   condition if it is irreducible and if, in addition, either the leading
   part is not a constant multiple of a power of an irreducible
   polynomial, or the leading part is not equal to any $\lambda\sb
   0$-leading part. For example, the Diophantine equation $y\sp 4+y\sp
   3-2x\sp 2y\sp 2+xy+3x\sp 3+x-5=0$ is irreducible, and hence it has
   only finitely many solutions by Runge's theorem. An example of an
   irreducible Diophantine equation not covered by Runge's theoem is
   $y\sp 4+xy\sp 2-2x\sp 3-18=0$. In theory, a bound of the form
   $\max(\vert x\vert ,\vert y\vert )<B$ would exist for the solutions
   $x,y$ to those Diophantine equations satisfying Runge's condition. In
   the paper of the reviewer and Straus, such a bound was actually
   computed by employing methods similar to those of Runge, namely, by
   considering the Puiseux series expansions of the algebraic function
   $y$ defined by $F(x,y)=0$. The computed quantity $B$ depends only on
   $h=\max\vert a\sb {ij}\vert $ (the height of the polynomial) and the
   degrees $d\sb 1,d\sb 2$. T. Skolem [Norsk. Mat. Forenings Skrifter
   Ser. I 1922, no. 10, 12 pp.; Jbuch 48, 139] gave another proof of
   Runge's theorem based on elimination theory. The authors use Skolem's
   approach to compute such a bound for Runge's equation.
   
                       Reviewed by David Lee Hilliker
   © Copyright American Mathematical Society 1994, 1999