Polynomial Parametrization of the Solutions of Certain Systems of Diophantine Equations

Abstract.

Let $f_1, f_2, \ldots , f_k \in {\mathbb {Z}}[X_0, X_1, \ldots , X_N]$ be non-constant homogeneous polynomials which define a projective variety V over $\mathbb {Q}$. Under the hypothesis that, for some $n \in \mathbb {N}$, there is a surjective morphism $\varphi: \mathbb {P}^n_\mathbb {Q} \rightarrow V$, we show that all integral solutions of the system of Diophantine equations f₁ = 0, . . . , f _k  = 0 (outside some exceptional set) can be parametrized by a single k-tuple of integer-valued polynomials. This result only depends on φ, but not on the embedding given by f₁, f₂, . . . , f _k . If, in particular, φ is a normalization of V, then the exceptional set is really small.