Polynomial Diophantine systems

On Diophantine Sets over Polynomial Rings

We prove that the recursively enumerable relations over a polynomial ring R[t], where R is the ring of integers in a totally real number field, are exactly the Diophantine relations over R[t].

Some polynomial formulas for Diophantine quadruples

The Greek mathematician Diophantus of Alexandria studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: 1 16 , 33 16 , 17 4 , 105 16 (see [4]). Fermat obtained four positive integers satisfying the condition of the problem above: 1, 3, 8, 120. For example, 3 · 120+1 = 1...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

A Polynomial Time Algorithm for Diophantine Equations in One Variable

We show that the integer roots of of a univariate polynomial with integer coe cients can be computed in polynomial time This re sult holds for the classical i e Turing model of computation and a sparse representation of polynomials i e coe cients and exponents are written in binary and only nonzero monomials are represented

Systems of quadratic diophantine inequalities

has a nonzero integer solution for every > 0. If some Qi is rational and is small enough then for x ∈ Zs the inequality |Qi(x)| < is equivalent to the equation Qi(x) = 0. Hence if all forms are rational then for sufficiently small the system (1.1) reduces to a system of equations. In this case W. Schmidt [10] proved the following result. Recall that the real pencil generated by the forms Q1, . ...