An Elementary Algorithm for Solving a Diophantine Equation of Degree Fourth with Runge’s Condition

We propose an elementary algorithm for solving the diophantine equation (p(x; y) + a1x + b1y)(p(x; y) + a2x + b2y)- dp(x; y)- a3x - b3y -c = 0 ( *) of degree fourth, where p(x; y) denotes an irreducible quadratic form of positive discriminant and (a1; b1) ̸= (a2; b2). The last condition guarantees that the equation ( ) can be solved using the well known Runge’s method, but we prefer to avoid the use of any power series that leads to upper bounds for solutions useless for a computer implementation.

First-order and counting theories of ω-automatic structures

The logic extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the set C”). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are many elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.