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.


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


Full Text: DOI arXiv


[1] Abbott, J.; Bigatti, A.; Kreuzer, M.; Robbiano, L., Computing ideals of points, J. Symbolic Comput., 30, 4, 341-356 (2000), MR MR1784266 (2001j:13026) · Zbl 0977.13011
[2] Abbott, J.; Kreuzer, M.; Robbiano, L., Computing zero-dimensional schemes, J. Symbolic Comput., 39, 1, 31-49 (2005), MR MR2168239 (2006g:13050) · Zbl 1120.13027
[3] Alonso, María Emilia; Marinari, Maria Grazia; Mora, Teo, The big mother of all dualities: Möller algorithm, Comm. Algebra, 31, 2, 783-818 (2003), MR MR1968924 (2004b:13029) · Zbl 1056.13015
[4] Alonso, María Emilia; Marinari, Maria Grazia; Mora, Teo, The big mother of all dualities. II. Macaulay bases, Appl. Algebra Engrg. Comm. Comput., 17, 6, 409-451 (2006), MR MR2270332 (2008d:13036) · Zbl 1114.13018
[5] Cox, David A.; Little, John; O’Shea, Donal, Ideals, varieties, and algorithms, (An Introduction to Computational Algebraic Geometry and Commutative Algebra. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergrad. Texts Math. (1997), Springer-Verlag: Springer-Verlag New York), MR MR1417938 (97h:13024) · Zbl 1335.13001
[6] Cox, David A.; Little, John; O’Shea, Donal, Using Algebraic Geometry, Grad. Texts in Math., vol. 185 (2005), Springer-Verlag: Springer-Verlag New York, MR MR2122859 (2005i:13037) · Zbl 1079.13017
[7] Griffiths, Phillip; Harris, Joseph, Principles of Algebraic Geometry, Pure Appl. Math. (1978), Wiley-Interscience [John Wiley & Sons]: Wiley-Interscience [John Wiley & Sons] New York, MR MR507725 (80b:14001) · Zbl 0408.14001
[8] Greuel, G.-M.; Pfister, G.; Schönemann, H., Singular 3.0, A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005
[9] Lederer, M., The vanishing ideal of a finite set of closed points in affine space, J. Pure Appl. Algebra, 212, 1116-1133 (2008) · Zbl 1137.13017
[10] Möller, H. M.; Buchberger, B., The construction of multivariate polynomials with preassigned zeros, (Computer Algebra (Marseille, 1982). Computer Algebra (Marseille, 1982), Lecture Notes in Comput. Sci., vol. 144 (1982), Springer-Verlag: Springer-Verlag Berlin), 24-31, MR MR680050 (84b:12003) · Zbl 0549.68026
[11] Marinari, M. G.; Möller, H. M.; Mora, T., Gröbner bases of ideals defined by functionals with an application to ideals of projective points, Appl. Algebra Engrg. Comm. Comput., 4, 2, 103-145 (1993), MR MR1223853 (94g:13019) · Zbl 0785.13009
[12] Sturmfels, Bernd, Gröbner Bases and Convex Polytopes, Univ. Lecture Ser., vol. 8 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, MR MR1363949 (97b:13034) · Zbl 0856.13020
[13] Wibmer, Michael, Gröbner bases for families of affine or projective schemes, J. Symbolic Comput., 42, 8, 803-834 (2007), MR MR2345838 · Zbl 1134.13025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.