Hilbert’s tenth problem over the rationals

Could one computer procedure decide whether any polynomial equation with integer coefficients has a solution made of fractions, and always finish? Coefficients are the fixed numbers in the equation’s terms; integers include negative numbers and zero as well as positive whole numbers. For example, $2x-1=0$ has coefficients 2 and −1 and the fractional solution $x=1/2$, while $x^2+1=0$ has no rational solution because a square cannot equal −1. Trying possible fractions can eventually find a solution if one exists. The challenge is also to finish with a correct 'no' when none exists.[1] [2]

The problem asks for a universal terminating test for rational solutions of polynomial equations. Given $f\in\mathbb{Z}[x_1,\ldots,x_m]$, with any finite number of variables, decide whether

$$f(q_1,\ldots,q_m)=0\quad\text{for some }(q_1,\ldots,q_m)\in\mathbb{Q}^{m}.$$

Here $\mathbb{Z}$ denotes the integers and $\mathbb{Q}$ the rational numbers. A decision algorithm must halt on inputs with solutions and inputs without them. Systematically enumerating rational tuples finds a solution if there is one, but supplies no general stopping rule in the other case. Undecidability over the integers does not transfer merely by allowing more candidate values.[1] [2]

Neither a universal algorithm nor an unconditional impossibility proof is known.[1] [2]

The rational case remains open. Integer problems and richer logical formulas do not settle this single-polynomial rational-solution question.[1] [2]

There is a difference between failing to find a solution and establishing that none exists. A universal procedure would bridge that gap for every equation of this kind. An impossibility proof would instead show a fundamental limit of computation, even when time and memory are not bounded in advance.[1] [2]

A resolution would place a precise boundary on algorithmic reasoning about ordinary rational equations. A positive answer would provide a terminating procedure without necessarily making it efficient. A negative answer would rule out every such universal procedure. One route toward the latter is to encode integer solvability using polynomial equations over the rationals, but the required encoding is itself unproved.[1] [2]