Projects

Re-Embeddings of Affine Varieties

Together with Dr. Le Ngoc Long (University of Passau and Hue University of Education) and Prof. Lorenzo Robbiano (Università di Genova), Prof. Kreuzer studies the efficient computation of re-embeddings of embedded affine varieties into lower dimensional affine spaces. They have been developing the method of Z-separating re-embeddings which allows to find and calculate such re-embeddings without the (potentially expensive) use of Gröbner bases.

Research papers:

• [1] M. Kreuzer, Le Ngoc Long and L. Robbiano, Cotangent spaces and separating re-embeddings, J. Algebra Appl. 21 (2022), DOI:10.48550/arXiv.2010.08378
• [2] M. Kreuzer, Le Ngoc Long and L. Robbiano, Restricted Gröbner fans and re-embeddings of affine algebras, São Paulo J. Math. Sci. (2022), 26 pages, DOI:10.1007/s40863-022-00324-w
• [3] M. Kreuzer, Le Ngoc Long and L. Robbiano, Optimal re-embeddings of border basis schemes, preprint 2022, 49 pages, available at arXiv:2207.08115 [math. AC]
• [4] M. Kreuzer, Le Ngoc Long and L. Robbiano, Re-embeddings of affine algebras via Gröbner fans of linear ideals, preprint 2023, 21 pages (submitted)

The following CoCoA code for paper [4] was implemented by Dr. Le Ngoc Long:

ApCoCoA package: GFanLin.cpkg5
Some functions in this package use the program GLPK which is currently supported only in the Linux version of ApCoCoA

ApCoCoA package (without GLPK): GFanLinear.cpkg5

Source code for the examples in [4]: GFanLinear-Examples.cocoa5

Subschemes of the Border Basis Scheme

In a series of papers, Dr. Le Ngoc Long (University of Passau and Hue University), Prof. Lorenzo Robbiano (Università di Genova) and Prof. Martin Kreuzer develop algorithms to check 0-dimenisonal schemes for certain algebraic and geometric properties. Furthermore, they derive algorithms to compute the locus of all schemes having a particular property inside the moduli space given by the border basis scheme.

Research Papers:

• [1] M. Kreuzer, Le Ngoc Long and L. Robbiano, On the Cayley-Bacharach property, Comm. Algebra 47 (2019), 328-354, DOI:10.1080/00927872.2018.1476525
• [2] M. Kreuzer, L.N. Long and L. Robbiano, Algorithms for checking zero-dimensional complete intersections, J. Commut. Algebra (to appear), available at arXiv:1903.09563 [math.AC]
• [3] M. Kreuzer, Le N. Long and L. Robbiano, Computing subschemes of the border basis scheme, Preprint 2019, available at arXiv:1910.09426 [math.AC]

The following CoCoA code was implemented by Dr. Le Ngoc Long:

CoCoA-5 package: BBS_subschemes.cpkg

Source code for the examples in [3]: BBS_examples.cocoa5

Algebraic Fault Attacks

Details can be found on the projects website.

ApCoCoA

In this software project the computeralgebra framework ApCoCoA is developed.

ApCoCoA