Effective Computation of Radical of Ideals and Its Application to Invariant Theory

The most expensive part of the known algorithms in the calculation of primary fundamental invariants (of rings of polynomial invariants of finite linear groups over an arbitrary field) is the computation of the radicals of complete intersection ideals. Thus, in this paper, we develop effective methods for such calculation. For this purpose, we introduce first a new notion of genericity (called

D-quasi stable position

) and exhibit a novel

deterministic

algorithm to put an ideal in Nœther position (we show that this new notion of genericity is equivalent to Nœther position). Then, we use this algorithm and also the algorithm due to Krick and Logar (to compute radicals of ideals) to present an efficient algorithm to calculate the radical of a complete intersection ideal. Furthermore, we apply this algorithm, to improve the classical methods of computing primary invariants which are based on radical computation. Finally, we have implemented in

Maple

the mentioned algorithms (to put an ideal in Nœther position, to compute the radical of ideals and also primary invariants) and compare the proposed algorithms, via a set of benchmarks, with the corresponding functions in

Maple

and

Magma

. The experiments we made seem to show that these first implementations are already more efficient than the corresponding functions of

Maple

and

Magma

.