Polynomial bounds in polynomial rings over fields (Q752093)

Let \(K\) be a field and let \(I\) be an ideal in the polynomial ring \(K[X] = K[X_1,\ldots,X_n]\). Suppose that \(I\) is generated by polynomials of (total) degree at most \(d\). The author uses non-standard set theory to give the following primality criterion. There is an integer \(\beta =\beta(n)\), independent of the field \(K\), so that \(I\) is prime if and only if the following holds: \(1\not\in I\), and for all \(f,g\in K[X]\) of degree \(\leq d^{\beta}\), if \(fg\in I\) then either \(f\in I\) or \(g\in I\). A principal result proved en route is that any system of linear equations over \(K[X]\) with coefficients of bounded degree \(d\) must have a solution (if any) of degree at most \(d^{\alpha}\), where \(\alpha\) depends only on \(n\).