Our fields of research are Computer Algebra and Computations of Commutative Algebra and their interactions with Differential Algebras. 

More precisely, we work on:

+ Zero-dimensional ideals of in a polynomial ring, Gröbner bases, and some relevant algebraic invariants  

+ Border bases and  border basis schemes parametrizing zero-dimensional schemes

+ Differential techniques for study of fat point schemes in (multi-) projective spaces

+ The problem of re-embedding of an affine schemes into a lower-dimensional affine space

+ The Gröbner fans, symbolic powers and differential powers of polynomial ideals

Publications

• Kähler differential algebras for 0-dimensional schemes and applications (TNK Linh), Thesis (Ph.D.)–Universität Passau, Passau, 2015.

• Various differents for 0-dimensional schemes and applications (LN Long), Thesis (Ph.D.)–Universität Passau, Passau, 2015.

• Kähler differentials and Kähler differents for fat point schemes (M. Kreuzer, T. N. K. Linh, and L. N. Long), J. Pure Appl. Algebra 219 (2015), no. 10, 4479–4509, DOI:10.1016/j.jpaa.2015.02.028 

• Characterizations of zero-dimensional complete intersections (M. Kreuzer and L. N. Long), Beitr. Algebra Geom. 58 (2017), no. 1, 93–129, DOI:10.1007/s13366-016-0311-9

• Kähler differential algebras for 0-dimensional schemes  (M. Kreuzer, T. N. K. Linh, and L. N. Long), J. Algebra 501 (2018), 255–284, DOI:10.1016/j.jalgebra.2017.12.023 

• A presentation of the Kähler differential module for a fat point scheme in ℙn1×…× ℙnk (TNK Linh, E Guardo, LN Long), ITM W. Conf. 20 (2018), 01007 DOI:10.1051/itmconf/20182001007

• On the Cayley-Bacharach property (M. Kreuzer, L. N. Long and L. Robbiano), Commun. Algebra 47 (2019), 328–354, DOI:10.1080/00927872.2018.1476525

• The Dedekind different of a Cayley–Bacharach scheme (M. Kreuzer, T. N. K. Linh, and L. N. Long), J. Algebra Appl. 18 (02) (2019), 1950027, DOI:10.1142/S0219498819500270 

• On the regularity of the monomial point of a border basis scheme (M. Kreuzer, B. Sipal, and L. N. Long), Beitr. Algebra Geom. 61 (2020), no. 3, 515–532, DOI:10.1007/s13366-019-00482-7 

• An application of liaison theory to 0-dimensional schemes (M. Kreuzer, T. N. K. Linh, L. N. Long and N. C. Tu), Taiwanese J. Math. 24 (2020), 553-573, DOI:10.11650/tjm/190710 

• Computing subschemes of the border basis scheme (M. Kreuzer, L. N. Long and L. Robbiano), Internat. J. Algebra Comput. 30 (2020), 1671-1716, DOI:10.1142/S0218196720500599 

• Kähler differentials of fat point schemes in P1 × P1 (E. Guardo, M. Kreuzer, T. N. K. Linh and L. N. Long), J. Commut. Algebra 13(2) (2021), 179-207, DOI:10.1216/jca.2021.13.179 

• Hilbert polynomials of Kähler differential modules for fat point schemes (M. Kreuzer, T. N. K. Linh and L. N. Long), Acta Math. Vietnam. 46 (3) (2021), 441–455, DOI:10.1007/s40306-021-00432-3 

• Algorithms for checking zero-dimensional complete intersections (M. Kreuzer, L. N. Long and L. Robbiano), J. Commut. Algebra 14 (2022), 61-76, DOI:10.1216/jca.2022.14.61 

• Cotangent spaces and separating re-embedding (M. Kreuzer, L. N. Long and L. Robbiano), J. Algebra Appl. 21 (2022), 2250188, DOI:10.1142/S0219498822501882 

• The Kähler different for a set of points in Pm × Pn (N. T. Hoa, T. N. K. Linh, L. N. Long, N. T. P. Nhi, P. T. T. Nhan), Bull. Korean Math. Soc. 59 (2022), 929-949,  DOI: 10.4134/BKMS.b210544

• Restricted Gröbner fans and re-embeddings of affine algebras (M. Kreuzer, L. N. Long and L. Robbiano), Sao Paulo J. Math. Sci. 17 (2023), 242–267, DOI:10.1007/s40863-022-00324-w 

• The Kähler different of a 0-dimensional scheme (L. N. Long). J. Algebra Appl. 23 (2024), No. 03, 2450056, DOI:10.1142/S0219498824500567 

• Zero-dimensional schemes and their moduli spaces (LN Long), Habilitation Thesis –Universität Passau, Passau, 2024.

• Re-embeddings of affine algebras via Gröbner fans of linear ideals  (M. Kreuzer, L. N. Long and L. Robbiano),  Beitr. Algebra Geom. 65 (2024), 827–851, DOI:10.1007/s13366-024-00733-2 

• Differential theory of zero-dimensional schemes (M. Kreuzer, T. N. K. Linh and L. N. Long), J. Pure Appl. Algebra 229 (2025),No. 1, 107815 , DOI:10.1016/j.jpaa.2024.107815  

• Some line and conic arangements and their Waldschmidt constants (T.D. Huynh, T. N. K. Linh and L. N. Long), J. Korean Math. Soc. (to appear 2025),  DOI: 10.4134/JKMS.j240438

• Efficent checking of separating re-embeddings (B. Andraschko, M. Kreuzer, L. N. Long), Math. Comput. Sci. 19 (9) (2025), DOI: 10.1007/s11786-025-00616-2

National Publications

• Cohomology degrees and reduction indices (TNK Linh), Journal of Science - Hue University of Education 2 (2007), 1859-1612.

• The Dedekind different for a uniform set of points in Pn (TNK Linh, LN Long), Hue University Journal of Science: Natural Science 131 (1B) (2022), 81-91, DOI:10.26459/hueunijns.v131i1B.6563

• Kähler differential modules and configurations of points in P2 (TNK Linh, LN Long),  Dalat University Journal of Science (2022), 48-60, DOI:10.37569/DalatUniversity.12.2.887(2022)

• The Gorenstein singularities of a 0-dimensional uniform scheme in Pn (with NT Hoa, NTP Nhi, PTT Nhan, TLD Tra), Journal of Science - Hue University of Education 3(71), (2024).

• Note on Restricted Gröbner Fans (TNK Linh, LN Long), Journal of Science - Hue University of Education 3(75) (2025).

Preprints

• Computing the Dedekind different of zero-dimensional schemes, preprint 2018.

• Optimal re-embedings of border basis schemes (with M. Kreuzer, L. Robbiano), preprint (2022), 49 pages.

• The de Rham cohomology of zero-dimensional schemes (with M. Kreuzer), preprint (2024), 15 pages.

• The canonical exact sequence of differential modules for 0-dimensional schemes, preprint 2025.

Ref. and Indices

• Hue-Uni-LNL - TNKL

• G-Scholar LNL - TNKL

• Researchgate LNL - TNKL

• Orcid-Number: LNL 0000-0001-7840-2315 - TNKL 0000-0001-7087-5396

• Scopus ID: LNL 57193239729  -  TNKL 57201367193

Books and Lecture Notes

• Computer Algebra (LN Long), University of Passau, Lecture Note, 2024.

• Linear Algebra I (TNK Linh, PD Dong, LN Long, DH Tuan: English and Vietnamese Versions), University of Education, Hue University,  2023.

• Elementary Number Theory (TNK Linh, PD Dong, LN Long, DH Tuan), University of Education, Hue University, Preparation.

• Coding Theory (LN Long), University of Passau, Lecture Note, 2022.

• Principle of Mathematics 1 & 2, University of Education, Hue University, Lecture Note 2024.

• Galois Theory (LN Long), University of Education, Hue University,  Lecture Note, 2020.

Several Talks

• Computational aspect of the uniformity property of finite sets of points , Workshop, Hoi An, 17/08/2025

• Hilbert functions for  zero-dimensional schemes, Workshop, Hoi An, 17/08/2025

• Computational Aspect of Re-embeddings of Affine Algebra, CoCoA school, 13-18/07/2025

• Differential methods for zero-dimensional schemes, Workshop, Hanoi, 28/03/2025

• Differential techniques for locally Gorenstein schemes, International conference, Tuy Hoa, 07/07/2024