Rational solutions of first-order algebraic ordinary difference equations (Q2307436)

The paper deals with the strong rational general solutions \(s\in \mathbb{K}(x,c)\setminus \mathbb{K}(x)\), \(c\) being a transcendental constant, for the algebraic ordinary difference equation (AO\(\Delta\)E) \[ F(x,y(x),y(x+1))=0,\tag{1} \] where \(F\in \mathbb{K}[x,y,z]\) is an irreducible polynomial, and \(\mathbb{K}\) is an algebraically closed field of characteristic zero. The authors prove that if the difference equation (1) admits a strong rational general solution, then its corresponding algebraic curve defined by \(F(x,y,z)=0\) is of genus zero. They also show that there is a one-to-one correspondence between the strong rational general solutions of (1) and those of the associated separable difference equation. For an autonomous first-order AO\(\Delta\)E, the authors give a bound for the degrees of rational solutions of its associated separable difference equation, and then they present a complete algorithm for computing rational solutions of autonomous first-order AO\(\Delta\)Es.