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Hyperplane sections, Gröbner bases, and Hough transforms. (English) Zbl 1311.13041

This paper is about hyperplane sections of affine schemes. It is shown that under certain assumptions a Gröbner bases of a scheme is closely related to Gröbner bases of the scheme’s hyperplane sections (Theorems 1.7 and 1.10). This was known in the homogeneous case, and this paper deals with the inhomogeneous case. It is shown how these results can be used to reconstruct schemes from enough hyperplane sections. This technique may have applications for example in implicitization (which can be much easier in hyperplane sections). The second part of the paper extends the results to parametrized families of schemes. This, in turn, may have applications in tomography, where one may want to reconstruct the shape of an organ from MRI data.

MSC:

13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14D06 Fibrations, degenerations in algebraic geometry
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
92C55 Biomedical imaging and signal processing
44A15 Special integral transforms (Legendre, Hilbert, etc.)

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References:

[1] CoCoA Team, CoCoA: a system for doing computations in commutative algebra, available at
[2] P.V.C. Hough, Method and means for recognizing complex patterns, US Patent 3069654, December 18, 1962.
[3] Kreuzer, M.; Robbiano, L., Computational Commutative Algebra 1 (2000), Springer: Springer Heidelberg · Zbl 0956.13008
[4] Kreuzer, M.; Robbiano, L., Computational Commutative Algebra 2 (2005), Springer: Springer Heidelberg · Zbl 1090.13021
[5] Beltrametti, M. C.; Robbiano, L., An algebraic approach to Hough transforms, J. Algebra, 371, 669-681 (2012) · Zbl 1295.13037
[6] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics (1995), Springer · Zbl 0819.13001
[7] Mora, T.; Robbiano, L., The Gröbner fan of an ideal, J. Symb. Comput., 6, 183-208 (1988) · Zbl 0668.13017
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