First off, I know that there's no general algorithm for determining if there's a solution to a general Diophantine equation, much less a system.


However, I'm wondering if there is an algorithm for solving a Diophantine system of linear and quadratic equations? In fact, I have a system which is "sparse" in some sense (the linear equations are all the sum of three variables equals a number, and all the same number, and the quadratics aren't much worse).


If so, then can it be extended to the case of countably many variables?



Algorithms for Diophantine Systems
No. Given any set of diophantine equations $f\_1(z\_1, \ldots, z\_n) = \ldots = f\_m(z\_1, \ldots, z\_n)=0$, we can rewrite in terms of linear equations and quadratics. Create a new variable $w\_{k\_1 \cdots k\_n}$ for each monomial $z\_1^{k\_1} \cdots z\_n^{k\_n}$ which occurs in the $f$'s, or which divides any monomial which occurs in the $f$'s. Turn each $f$ into a linear equation: For example, $x^3 y^2 + 7 x^2 y=5$ becomes $w\_{32} + 7 w\_{21} = 5$. Then create quadratic equations $z\_i w\_{k\_1 \cdots k\_i \cdots k\_n} = w\_{k\_1 \cdots (k\_i +1) \cdots k\_n}$. For example, $x w\_{22} = w\_{32}.$ This shows that the solvability of Diophantine equations is equivalent to that of Diophantine equations of degree $\leq 2$.


I'll also mention a very concrete case. The intersection of two quadrics in $\mathbb{P}^3$ is a genus $1$ curve. To my knowledge, no algorithm is known to test for the existence of rational points even in this case. (But my knowledge is not very large.) 


