TITLE: Algorithms for Diophantine Systems
QUESTION [6 upvotes]: 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?

REPLY [14 votes]: 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.)