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Nœther bases and their applications. (English) Zbl 1430.13045

It is well known that, in generic coordinates, the initial ideal \(J\) of a polynomial ideal \(I\subset R:=\mathcal K[x_1,\dots,x_n]\) over a field \(\mathcal K\) with respect to a term order acquires special properties that, in some cases, can be also described by a combinatorial behaviour. For example, \(J\) could be quasi-stable. Such a situation is also significant in term order-free contexts in which the monomial ideal \(J\) is not necessarily an initial ideal of \(I\) but characterizes \(I\) by the condition that the sum between \(I\) and the vector space generated by the terms outside \(J\) is direct and covers all the polynomial ring. This topic is important in Computational Algebra and has possible relevant consequences in Commutative Algebra and Algebraic Geometry.
In the paper under review, a term order is supposed to be given. In this case it is interesting to study if the monomial ideal \(J\) is related to involutive divisions, as it happens for quasi-stable ideals. The possible combinatorial nature of the ideal \(J\) can be determined by the existence of special finite sets of generators, like Pommaret bases for quasi-stable ideals. The existence of such special bases for \(J\) reflects the existence of special bases for the ideal \(I\).
Here, the notion of \(D\)-Noether basis for a homogeneous polynomial ideal \(I\) is introduced (see Definition 3.8) and its existence is investigated, where \(D\) is the Krull dimension of the quotient of the polynomial ring \(R\), with \(\mathcal K\) an infinite field, over \(I\).
Recall that \(I\) is in Noether position if \(\mathcal K[x_{n-D+1},\dots, x_n] \hookrightarrow R/I\) is an integral ring extension, which is called a Noether normalization. There always exists a suitable linear change of variables up to which \(I\) is in Noether position. The condition to be in Noether position is weaker than to be in quasi-stable position and some studies on this topic have been already presented in several papers with different aims (see the References).
The authors of the paper under review show that a homogeneous ideal \(I\) is in Noether position if and only if it has a finite \(D\)-Noether basis if and only if \(J\) is weakly \(D\)-quasi stable (see Lemma 2.13, Definition 2.14, Theorems 2.15 and 3.11, Corollary 3.12). These interesting results are described meanwhile the authors also introduce the \(D\)-Noether division proving that this is an involutive division (Proposition 3.3) and a Noether basis is an involutive basis. Although this division is not Noetherian, the authors also describe deterministic algorithms to transform an ideal in Noether position and to compute a Noether basis of the transformed ideal.

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
68W30 Symbolic computation and algebraic computation
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References:

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