An algorithm for solving a family of fourth-degree Diophantine equations that satisfy Runge's condition (Q2027843)

In this paper, the Diophantine equation \[f(x,y)= x(ax^3+ bx^2 y+ cxy^2 dy^3)+xg(x,y)+h(y) = 0,\] where \[g(x,y) = p_0x^2+(p_1y+p_2)x+p_3y^2+p_4y+p_5, \quad h(y) = Ay^2+By+C\] and the coefficients of \(f(x,y)\) are integers with \(d\ne 0\), is studied. For the determination of the solutions of this equation the so-called \textit{Runge's method} can be applied. This paper proposes an algorithmic implementation of an elementary version of this method for solving the above equation. The proposed algorithm differs from similar algorithms as it requires solving a number of fourth-degree equations in one unknown over the integers which must be taken into account to correctly estimate the complexity of the algorithm. The optimized version algorithm is implemented in the PARI/GP computer algebra system.