Noether normalization guided by monomial cone decompositions
Noether normalization is a cornerstone in computational commutative algebra. In the last two decades, many algorithms based on Gröbner bases and random changes of variables have been introduced by several authors. In this paper, the author uses Janet bases and the monomial cone decompositions they define to produce an effective implementation of a general algorithm for Noether normalization outlined by Vasconcelos in 1998 [\textit{W. V. Vasconcelos}, Computational methods of commutative algebra and algebraic geometry. Algorithms and Computation in Mathematics 2. Berlin: Springer (1988; Zbl 0896.13021)]. The main advantages in this novel use of Janet bases for Noether normalization are the detection of \textit{sparse} coordinate transformations and the ability to decide whether an ideal contains a monic polynomial in a given variable without computing elimination ideals (which is usually done via Gröbner bases). A suitable choice of coordinate transformation is crucial in this procedure to avoid computational bottlenecks arising from the fact that arbitrary transformations significantly increase the size of coefficients and the number of monomials in a given polynomial. Hence the procedure outlined in this paper has the promise of being significantly faster than most previous approaches. This is confirmed by an experiment performed by the author comparing his implementation against standard procedures found in some of the main computer algebra systems.
scientific article; zbMATH DE number 5603610
scientific article