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Tài Huy Hà

Publications and source records attributed to Tài Huy Hà.

15 recordsLinked to original sources
arXiv2026

Edge ideals of gapfree graphs with matching number $3$

We prove that if $G$ is a gapfree graph with matching number $3$, then every power $I(G)^q$, $q\geq2$, of its edge ideal has linear quotients, and hence a linear resolution. Our approach combines graph reductions based on clique domination of gapfree graphs and matching number with operations preserving linear quotients. We reduce the problem to $26$ non-co-chordal graphs on at most seven vertices. A new preservation theorem for gapfree edge addition, together with known results on duplication and gapfree expansion, handles all but one of these graphs. For the remaining graph on seven vertices, we construct a linear quotients ordering for the square and extend it recursively to every higher power.

math.AC↗
arXiv2026

Cohen-Macaulay higher conormal and Kähler differential modules of squarefree monomial ideals

Let $S=k[x_1,\ldots,x_n]$ and let $I=I_Δ\subsetneq S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I/I^q$, but more generally the shifted quotients $I^r/I^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}/I^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge3$, the module $I^r/I^q$ is Cohen--Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}/I^{(q)}$ is Cohen--Macaulay if and only if $Δ$ is a matroid, where loops are allowed. Thus, the Cohen--Macaulayness of a single nonexceptional window forces the Cohen--Macaulayness of every window in the corresponding filtration. The unique exceptional pair is $(r,q)=(1,2)$: at this conormal level, we show that the Cohen--Macaulayness of $I/I^2$ forces $I^2=I^{(2)}$, and hence $I/I^{(2)}$ is Cohen--Macaulay.

math.AC↗
arXiv2026

Comparing v-numbers of symbolic and ordinary powers of squarefree monomial ideals

Let $I$ be a squarefree monomial ideal, and let $d$ denote the maximum degree of a minimal generator of $I$. We prove that \[ v(I^t)\le v(I^{(t)})+(t-1)(d-1) \] for all $t\ge1$, where $I^{(t)}$ denotes the $t$-th symbolic power of $I$. In particular, this bound does not depend on the number of variables in the ambient polynomial ring. On the other hand, for every fixed exponent $t\ge2$, the difference \[ v(I^{(t)})-v(I^t) \] can be arbitrarily large. Finally, we determine the $v$-numbers of the ordinary and symbolic powers of edge ideals of paths and cycles.

math.AC↗
arXiv2026

Polynomial interpolation and the Waldschmidt constant of points in projective space

We establish Chudnovsky's Conjecture and, more generally, Demailly's Conjecture for any finite set of points in $\mathbb{P}^n_k$, where $k$ is an algebraically closed field of characteristic 0. The key idea of passing to positive characteristic, as well as the broad plan of the proof, was suggested by ChatGPT-5.6 Sol. All mathematical arguments and proofs in this paper were developed, written, and verified by the authors.

math.AG↗
arXiv2026

Regular sequences of linear forms on monomial ideals

In this paper we establish a means of using the combinatorics associated to a general monomial ideal $I$ in a polynomial ring $R$ to find a regular sequence of linear forms on $R/I$. The sequence of linear forms provides an effective lower bound on ${\rm{depth}}(R/I)$. When $I$ is the edge ideal of a graph, we provide conditions under which this bound is an equality, allowing the realization of the depth via a regular sequence of linear polynomials. In addition, we explicitly describe the minimal primes of $(I, f_1, \ldots, f_q)$, when $f_1, \ldots, f_q$ are homogeneous polynomials of degree one with pairwise disjoint support and $I$ is any monomial ideal. Finally, we propose a conjecture on the form of all associated primes of the ideal $(I, f_1, \ldots, f_q)$, when $I$ is the edge ideal of a graph and $f_i$ are disjoint stars on $I$.

math.AC↗
arXiv2026

Scarf complexes of graphs and their powers

Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs.

math.AC↗
arXiv2025

Multiplicity $=$ Volume formula and Newton non-degenerate ideals in regular local rings

We develop the notions of Newton non-degenerate (NND) ideals and Newton polyhedra for regular local rings. These concepts were first defined in the context of complex analysis. We show that the characterization of NND ideals via their integral closures known in the analytical setting extends to regular local rings. We use the limiting body $\mathcal{C}(\mathcal{I})$ associated to a graded family $\mathcal{I}$ of ideals to provide a new understanding of the celebrated "Multiplicity $=$ Volume" formula. Particularly, we prove that, for a Noetherian graded family $\mathcal{I}$ of $\mathfrak{m}$-primary ideals in a regular local ring $(R,\mathfrak{m})$ of dimension $d$, the equality $$e(\mathcal{I}) = d!\text{co-vol}_d(\mathcal{C}(\mathcal{I}))$$ holds if and only if $\mathcal{I}$ contains certain subfamily of NND ideals.

math.AC↗
arXiv2025

Rational powers, invariant ideals, and the summation formula

We provide explicit descriptions for the rational powers and Rees valuations of several classes of ideals invariant under natural actions of tori and products of general linear groups, in terms of polyhedra and lattice points. This allows us to show that a version of Mustaţă-Takagi's summation formula for multiplier ideals also holds for the rational powers of these ideals. Moreover, for arbitrary ideals in normal domains that are finitely generated over algebraically closed fields, we prove a weaker version of this formula that holds for sufficiently large rational numbers.

math.AC↗
arXiv2025

Vertex connectivity of chordal graphs

Let $G$ be a finite graph and $κ(G)$ the vertex connectivity of $G$. A chordal graph $G$ is called chordal$^*$ if no vertex of $G$ is adjacent to all other vertices of $G$. Using the syzygy theory in commutative algebra, it is proved that every chordal$^*$ graph $G$ on $n$ vertices satisfies $κ(G) \leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$. Furthermore, given an integer $0 \leq κ\leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$, a chordal$^*$ graph $G$ on $n$ vertices satisfying $κ(G) = κ$ is constructed.

math.CO↗
arXiv2024

Gapfree graphs and powers of edge ideals with linear quotients

Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$.

math.AC↗
arXiv2024

Minimal cellular resolutions of powers of graphs

Let $G$ be a connected graph and let $I(G)$ denote its edge ideal. We classify when $I(G)^n$, for $n \ge 1$, admits a minimal Lyubeznik resolution. We also give a characterization for when $I(G)^n$ is bridge-friendly, which, in turn, implies that $I(G)^n$ has a minimal Barile-Macchia cellular resolution.

math.AC↗
arXiv2024

Resolution and Betti numbers of Vertex cover ideals

The vertex cover ideal $J(G)$ of a finite graph $G$ is studied. We characterize when a Cohen--Macaulay vertex cover ideal $J(G)$ has a Scarf minimal free resolution. Furthermore, by using both combinatorial and topological techniques, the graded Betti number $β_{i,i+j}(J(G))$, where $i$ and $j$ are the projective dimension and the regularity of $J(G)$, is computed, when $G$ is either a path or a cycle.

math.AC↗
arXiv2023

Resurgence number of graded families of ideals

We define the resurgence and asymptotic resurgence numbers associated to a pair of graded families of ideals in a Noetherian ring. These notions generalize the well-studied resurgence and asymptotic resurgence of an ideal in a polynomial ring. We examine when these invariant are finite and rational. We investigate situations where these invariant can be computed via Rees valuations or realized as actual limits of well-defined sequences. We study how the asymptotic resurgence changes when a family is replaced by its integral closure. Many examples are given to illustrate that whether or not known properties of resurgence and asymptotic resurgence of an ideal would extend to that of a pair of graded families of ideals generally depends on the Noetherian property and finite generation of the Rees algebras of these families.

math.AC↗
arXiv2022

Regular Sequences On Squares Of Monomial Ideals

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular sequences on $R/I^2$. We then obtain a criterion for when $\depth R/I^2>1$ and a lower bound on $\depth R/I^2$.

math.AC↗