Powers of edge ideals and their homological invariants. Together with Nguyen Cong Minh, Le Dinh Nam, Thieu Dinh Phong, and Phan Thi Thuy, we laid out a new approach toward understanding homological invariants of edge ideals of graphs. In subsequent work with N. C. Minh, we continue to provide further steps toward understanding the regularity of powers of edge ideals.
Linearity defect of matrices of linear forms. Recently, with Hop Nguyen, I compute the linearity defect of ideals of minors of 2 x e matrices of linear forms. The proof exhibits many Betti splittings and guides to many other problems connected to the regularity of the ideal of minors.
Regularity of symbolic powers of monomial ideals. I am interested in the explicit calculation of the regularity of symbolic powers of some monomial ideals.
Regularity and defining equations of monomial curves. I am interested in bounds for the regularity and the number of defining equations of monomial curves.
Regularity of squarefree monomial ideals with partial linear resolutions. Together with Long Dao, I study the problem of bounding the regularity of squarefree monomial ideals whose minimal free resolutions are k-step linear. We prove a sharp linear upper bound in the case of ideals which are 2-step linear.
Unmixed and sequentially Cohen-Macaulay skew tableau ideals, Journal of Algebraic Combinatorics 62(2025),26.
We associate a skew tableau ideal to each filling of a skew Ferrers diagram with positive integers. We classify all unmixed and sequentially Cohen-Macaulay skew tableau ideals. Consequently, we classify all Cohen-Macaulay, Buchsbaum, and generalized Cohen-Macaulay skew tableau ideals.
Cochordal zero divisor graphs and Betti numbers of their edge ideals, Communications in Algebra (2025).
We associate a sequence of positive integer, called type, to each cochordal graph and use it to compute all the Betti numbers of the associated edge ideal. We then classify all integer n for which the zero divisor graph of Z/nZ is cochordal and compute all the Betti numbers of these graphs.
Betti numbers of powers of path ideals of cycles, with S. Balanescu and M. Cimpoeas, Journal of Algebraic Combinatorics 61 (2025), 44.
We compute all the Betti numbers of powers of t-path ideals of n-cycles when t = n-1 or t = n-2. The computations utilize Betti splittings and pave the way for further computations of Betti numbers for powers of path ideals arising from other classes of graphs.
Regularity of normal Rees algebras of edge ideals of graphs, with T. Q. Hoa and C. H. Linh, Journal of Algebra 680 (2025), 1-11.
We define a new class of graphs, called Tutte-Berge graphs. We show that a graph whose Rees algebra of its edge ideal is normal has regularity equal to the matching number if and only if it is Tutte-Berge.
Depth and regularity of tableau ideals, with D. T. Hoang, Advances in Applied Mathematics 169 (2025), 102913.
We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.
Regularity of powers and symbolic powers of edge ideals of cubic circulant graphs, with N. T. Hang and M. H. Pham, Annals of Combinatorics (2025)
We compute the regularity of powers and symbolic powers of edge ideals of all cubic circulant graphs.
Multiplicity of powers of squarefree monomial ideals, with P. T. Thuy, Archiv der Mathematik 125 (2025), 9-15.
We provide a formula for the multiplicity of powers of an arbitrary squarefree monomial ideal based on its dimension and the number of associated primes of maximal dimension. We then compute the multiplicity of powers of path ideals of cycles.
Projective dimension and regularity of 3-path ideals of unicyclic graphs, with N. T. Hang, Graphs and Combinatorics 41 (2025), 18.
We compute the projective dimension and regularity of 3-path ideals of arbitrary trees and unicyclic graphs.
Betti numbers of the tangent cones of monomial space curves, with N. P. H. Lan and N. C. Tu, Acta Mathematica Vietnamica 49 (2024), 347-365.
We describe a Grobner basis of the tangent cones for arbitrary monomial space curves. From that, we deduce a strong form of the conjecture of Herzog and Stamate bounding the Betti numbers of the tangent cones of monomial space curves.
The sequentially Cohen–Macaulay property of edge ideals of edge-weighted graphs, with L. T. K. Diem and N. C. Minh, Journal of Algebraic Combinatorics 60 (2024), 589-597.
Let G be a simple graph and w be a weight function on the edges of G. We prove that the edge ideal of the edge-weighted graph (G,w) is Cohen-Macaulay for all weight function w if and only if G is the disjoint union of complete graphs. We also prove that (G,w) is sequentially Cohen-Macaulay for all weight function w if and only if G is a Woodroofe graph, i.e., G does not have induced cycles other than triangles and pentagons. The proof utilizes associated radicals of I(G,w).
Depth of powers of edge ideals of Cohen-Macaulay trees, with N. T. Hang and T. T. Hien, Communications in Algebra 52 (2024), 5049-5060.
We prove that the depth of powers of edge ideals of Cohen-Macaulay trees drops down exactly one until it stabilizes.
Stable value of depth of symbolic powers of edge ideals of graphs, with N. C. Minh and T. N. Trung, Pacific Journal of Mathematics 329 (2024), 147-164.
Let G be a simple graph. We define the bipartite connectivity number of G, which is the minimum number of connected components in a maximal induced bipartite subgraph of G. We prove that the limit depth of symbolic powers of the edge ideal of G is at most the bipartite connectivity number of G. Additionally, we define a finer invariant called the restricted bipartite connectivity number of G and conjecture that the limit depth of the symbolic powers of the edge ideal of G is equal to this finer invariant. We verify our conjecture when G is a whisker graph of complete graphs. We further compute the depth of symbolic powers of edge ideals of cycles.
A characterization of Graphs whose small powers of their edge ideals have a linear free resolution, with N. C. Minh, Combinatorica 44 (2024), 337-353.
Let G be a simple graph. We prove that the second power of I(G) has a linear free resolution if and only if G is gap-free and has regularity at most 3. Similarly, we prove that the third power of I(G) has a linear free resolution if and only if G is gap-free and has regularity at most 4. We deduce these characterizations by computing the regularity of the second and third powers of edge ideals of arbitrary gap-free graphs.
Integral closure of powers of edge ideals and their regularity, with N. C. Minh, Journal of Algebra 609 (2022), 120-144.
We prove that the regularity of integral closure of powers of edge ideals is equal to that of regular powers for the third and fourth powers. We also compute the regularity of integral closure of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes. We further provide examples of edge ideals of graphs whose regularity of all powers depends on the characteristic of the base field.
Regularity of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes, with N. C. Minh, Mathematische Nachrichten 296 (2023), 3539-3558.
We give a formula for the regularity of powers and all intermediate ideals lying between regular powers and symbolic powers of Stanley-Reisner ideals of one-dimensional simplicial complexes.
Comparison between regularity of small symbolic powers and ordinary powers of an edge ideal, with N. C. Minh, L. D. Nam, T. D. Phong, and P. T. Thuy, Journal of Combinatorial Theory, Series A 190 (2022), 105621.
We lay out a framework for comparing the regularity of two monomial ideals via the study of degree complexes. Using this, we prove that the regularity of the second/third symbolic powers of edge ideals of graphs is equal to that of the second/third regular powers of the corresponding edge ideals.
Survey on Regularity of symbolic powers of an edge ideal, with N. C. Minh, In Peeva I. (eds) Commutative Algebra. Springer, Cham. (2022)
Let G be a simple graph and I its edge ideal. We survey recent results on the problem of computing the regularity of symbolic powers of I and its relation to the regularity of ordinary powers. We prove a rigidity property of the regularity for intermediate ideals lying between ordinary powers and symbolic powers for small powers. We then propose some related problems.
Algebraic invariants of projections of varieties and partial elimination ideals, with S. Kwak and Hop Nguyen, Journal of Algebra 586 (2021), 973-1013.
We give bounds on regularity and depth of an ideal I in terms of its partial elimination ideals.
Homological invariants of powers of fiber products, with Hop Nguyen, Acta Mathematica Vietnamica 44 (2019), 617–638.
Let I and J be ideals in polynomial rings R, S. Denote F the fiber product of I and J. We give a formula for homological invariants of F in terms of the corresponding invariants of I and J.
Products of ideals of linear forms in quadric hypersurfaces, with A. Conca and Hop Nguyen, Proceedings of the American Mathematical Society 147, no. 5 (2019), 1867–1880.
We prove that products of linear ideals in a quadric hypersurface have infinite free linear resolutions.
Powers of sums and their homological invariants , with Hop Nguyen, Journal of Pure and Applied Algebra 223 (2019), 3081–3111.
Let I and J be ideals in polynomial rings R, S. Denote P the sum of I and J. We give a formula for homological invariants of P in terms of the corresponding invariants of I and J.
Quantum steering with positive operator valued measures, with C. Nguyen, A. Milne, and S. Jevtic, Journal of Physics A: Mathematical and Theoretical 51 (35) (2018), 355302.
We show that, given a candidate for local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartite quantum state of finite dimension with POVMs can be formulated as a nesting problem of two convex objects. One consequence of this is the strengthening of the theorem that justifies choosing the LHS ensemble based on symmetry of the bipartite state. As a more practical application, we study the classic problem of the steerability of two-qubit Werner states with POVMs.
On the asymptotic behavior of the linearity defect, with Hop Nguyen, Nagoya Mathematical Journal 230 (2018), 35-47.
We prove that over a regular local ring, large enough powers of an ideal have a constant linearity defect.
Lower bounds on projective levels of complexes, with H. Altmann, E. Grifo, J. Montano, and W. Sanders, Journal of Algebra 491 (2017), 343-356.
We prove that the level of a perfect complex is bounded below by the largest gap in its homology. As a consequence, we deduce an improved version of the New Intersection Theorem for rings containing a field. We also obtain levels of Koszul complexes of various ideals.
Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outimes, with Chau Nguyen, Europhysics Letters 115 (2016), 10003
We define the critical radius of local model and prove that a two-qubit state is steerable if and only if the critical radius is smaller than 1. Using that we establish a conjecture of Jevtic et al on the steerability of T-states.
Non-separability and steerability of two-qubit states from the geometry of steering outcomes, with Chau Nguyen, Physical Review A 94 (2016), 012114
We prove that the non-separability and steerability of a two-qubit state are completely determined by the packability of the double cone of steering outcomes.
The Waldschmidt constant for squarefree monomial ideals, with C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, and A. Van Tuyl, Journal of Algebraic Combinatorics 44 (2016), 875-904.
We expressed the Waldschmidt constant of a squarefree monomial ideal in terms of the fractional chromatic number of a hypergraph associated to it. Moreover, we prove a Chudnovsky-like lower bound on the Waldschmidt constant, thus verifying a conjecture of Cooper-Embree-Ha-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals.
Regularity of products over quadratic hypersurfaces, with Hop Nguyen, Extended Abstracts Spring 2015, 129–133, Trends in Mathematics, Springer–Birkhäuser, Basel, 2016.
We prove that products of linear ideals over quadric hypersurfaces almost have linear resolutions.
Linearity defects of edge ideals and Fr\"oberg theorem, with Hop Nguyen, Journal of Algebraic Combinatorics 44 (2016), 165-199
We classify all graphs whose edge ideals have linearity defect at most one. They are weakly chordal graphs with induced matching number at most two. We further find formula for linearity defect of arbitrary weakly chordal graph, and of cycles. We also compute the linearity defect of the examples of Katzmann and Dalili-Kummini to show that the linearity defect of edge ideals depends on the characteristic of the base field. The later computation is done using our M2 package LinearityDefect.
Koszul determinantal rings and 2 x e matrices of linear forms, with Hop Nguyen and Phong Thieu, Michigan Mathematical Journal 64 (2015), 349-381.
We classify matrices of linear forms whose determinantal rings are Koszul. As an application, we classify linearly (and universally linearly) Koszul rational normal scrolls.
Regularity over homomorphisms and a Frobenius characterization of Koszul algebras, with Hop Nguyen, Journal of Algebra 429 (2015), 103-132.
Let k be an F-finite field of characteristic p. Let R be a standard graded algebra over k. We prove that R is Koszul if and only if there exists a finitely generated R-module M such that the regularity of M under the Frobenius action over R is finite.
Periodicity of Betti numbers of monomial curves, Journal of Algebra 418 (2014), 66-90
We prove the Herzog-Srinivasan conjecture saying that the Betti numbers of a shifted family of monomial curves are eventually periodic.
N6 property for third Veronese embeddings, Proceedings of the American Mathematical Society 143 (2015), 1897-1907
We prove that third Veronese embeddings of projective spaces satisfy property N6. This settles the Ottaviani-Paoletti conjecture in the case of third Veronese embeddings.
Fourth Veronese embeddings of projective spaces pdf
We prove that fourth Veronese embeddings of projective spaces satisfy property N9 verifying the Ottaviani-Paoletti conjecture in this case.
The Koszul property of pinched Veronese varieties arXiv (The proof has an error.)
A pinched Veronese variety is a projection of a Veronese embedding of projective space. We classify all Koszul pinched Veronese varieties.