Efficient algorithms for computing Noether normalization. In this paper, we provide first a new algorithm for testing whether a monomial ideal is in Noether position or not, without using its dimension, within a complexity which is quadratic in input size. Using this algorithm, we provide also a new algorithm to put an ideal in this position within an incremental (one variable after the other) random linear change of the last variables without using its dimension. We describe a modular (probabilistic) version of these algorithms for any ideal using the modular method used in [E. A. Arnold, “Modular algorithms for computing Gröbner bases”, J. Symb. Comput. 35, No. 4, 403–419 (2003; Zbl 1046.13018)] with some modifications. These algorithms have been implemented in the distributed library noether.lib [A. Hashemi, “noether.lib. A singular 3.0.3 distributed library for computing the nœther normalization” (2007)] of Singular, and we evaluate their performance via some examples.

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