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.
-  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
-  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
-  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]
-  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  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 : GFanLinear-Examples.cocoa5
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.
-  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
-  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]
-  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 : BBS_examples.cocoa5
Details can be found on the projects website.
In this software project the computeralgebra framework ApCoCoA is developed.