The complexity of computing the Hilbert polynomial of smooth complex projective varieties

Despite the impressive progress in the development of algebraic algorithms and computer algebra packages, the inherent computational complexity of even the most basic problems in algebraic geometry is still far from being understood. In [5] a systematic study of the inherent complexity for computing algebraic/topological quantities was launched with the goal of characterizing the complexity of various such problems by completeness results in a suitable hierarchy of complexity classes. We continue this study by investigating the complexity of computing the Hilbert polynomial of a complex projective variety V ⊆ P. This polynomial encodes important information about the variety V , like its dimension, degree and arithmetic genus. Several algorithms for computing the Hilbert polynomial were described by Mora and Möller [14] and Bayer and Stillmann [3]. The latter algorithm has been implemented in the computer algebra system Macaulay 2 and works quite well in practice. All these algorithms are based on the computation of Gröbner bases, which leads to bad upper complexity estimates. In fact, it follows from the work of Mayr and Mayer [13] that the problem of computing a Gröbner basis is exponential space complete. Both cardinality and maximal degree of a Gröbner basis might be doubly exponential in the number of variables (cf. [12]). It is generally believed that these bounds are quite pessimistic and that for problems with “nice” geometry single exponential upper bounds should hold for Gröbner bases. However, only few rigorous results are known in this direction [2,7,9,12]. Thus, currently, no upper bound better than exponential space is known for the computation of the Hilbert function or polynomial of a homogeneous ideal. Based on the lower bound on the homogeneous polynomial ideal membership problem in [12] we are able to show that the problem of computing the Hilbert polynomial is FPSPACE-hard, where FPSPACE denotes the