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Finite sets of \(d\)-planes in affine space. (English) Zbl 1191.14072

For a variety \(A\) in \(\mathbb{A}^n\) and a term order \(<\) on \(k[X_1,\dots,X_n]\), the author considers the set \(D(A)\subset\mathbb{N}^n\) consisting of all exponents of non-leading terms of the vanishing ideal \(I\) of \(A\). In the special case of a variety \(A\) whose irreducible components are \(d\)-dimensional affine subspaces of \(\mathbb{A}^n\), the geometric properties of \(A\) and the combinatorial properties of \(D(A)\) are related. For instance, the author shows that the number of irreducible components of \(A\) equals the number of so-called \(d\)-planes in \(D(A)\), where a \(d\)-plane is a subset of the form \(\gamma+\bigoplus_{j\in J}\mathbb{N} e_j\) with \(\#J= d\) and \(\gamma_j= 0\) for \(j\in J\). Furthermore, the author uses a kind of addition for sets of type \(D(A)\) and additional hypotheses to compute \(D(A)\) in terms of the sets \(D(A_\lambda)\), where \(\lambda\in K\) and \(A_\lambda= A\cup\{X_1=\lambda\}\). The paper is elementary in nature, clearly written and contains several examples.

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14Q99 Computational aspects in algebraic geometry

Software:

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References:

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