of a free module of dimension over if the coefficients over a basis of of a generating system of have a total degree at most , then there is a constant such that the degrees occurring in a generating system of the first syzygy module is at most The same bound applies for testing the membership to of an element of . On the other hand, there are examples where a double exponential degree necessarily occurs. However such examples are extremely rare, and this sets the question of an algorithm that is efficient when the output is not too large. At the present time, the best algorithms for computing syzygies are Gröbner basis algorithms. They allow the computation of the first syzygy module, and also, with almost no extra cost, all syzygies modules. Syzygies and regularity One might wonder which ring-theoretic property of causes the Hilbert syzygy theorem to hold. It turns out that this is regularity, which is an algebraic formulation of the fact that affine -space is a variety without singularities. In fact the following generalization holds: Let be a Noetherian ring. Then has finite global dimension if and only if is regular and the Krull dimension of is finite; in that case the global dimension of is equal to the Krull dimension. This result may be proven using Serre's theorem on regular local rings. See also Quillen–Suslin theorem Hilbert series and Hilbert polynomial References David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ; Commutative algebra Homological algebra Invariant theory Theorems in ring theory