On Puiseux series whose curves pass through an infinity of algebraic lattice points

An algorithm for computing the values of the ramification index in the Puiseux series expansions of an algebraic function

“…To illustrate the ideas, let us compute the ramification index of the algebraic function w given by w 3 -(1 + 3z)w 2 + 3z 2 w -z 3 + z = 0. Newton's polygon has only one vertex, namely, (1,3). Thus all branches are of degree 1.…”

“…The equation (14) for the leading coefficient is (d 0 -I) 3 = 0, so that d 0 = 1 with multiplicity 3. Condition (8) fails for w. The algebraic function v λ of z λ = z is given by w = z + zυ l9 and has the defining equation z 3 v\z 2 v\ -2z 2 v λ -z 2 + z = 0. All branches of υ λ are of degree -1/3.…”

Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem

“…We next genralize the Puiseux series theorem to curves which pass through infinitely many algebraically integral lattice points, subject to suitable restrictions (see also Hilliker and Straus [6]). We then obtain a corresponding generalization of Runge's Theorem for binary Diophantine equations over an algebraic number field.…”

Polynomial Bounds for the Solutions of a Class of Diophantine Equations

“…As far I know the result of Theorem 1 is the first example of a polynomial bound for the height of integer points of a class of algebraic curves defined over a number field bigger than Q. For the case that K=Q, Runge's method can be applied to a certain class of Diophantine equations, and in this case polynomial bounds are obtainable (see [4,5,6,20]). In the case that our method applies, we are able to improve on these bounds.…”