arXiv subjects

Tai Huy Ha

Publications and source records attributed to Tai Huy Ha.

8 recordsLinked to original sources

arXiv2026

Frobenius dilation of Hilbert function, Demailly's conjecture, and varieties of powers

We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem--Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha--Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of $V^k_{2,d}$ for all $k\ge3$ and $d\ge2$; in particular, it settles the ternary Waring problem for generic $k$-rank conjectured by Lundqvist--Oneto--Reznick--Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent $q$, any failure of Nicklasson's conjecture in three variables is confined to $2q-2$ consecutive degrees.

math.AC↗

arXiv2026

An algebraic theory of Lojasiewicz exponents

We develop a unified algebraic and valuative theory of Lojasiewicz exponents for pairs of graded families and filtrations of ideals. Within this framework, local Lojasiewicz exponents, gradient exponents, and exponents at infinity are all realized as asymptotic containment thresholds between filtrations, governed by integral closure. This reformulation shows that Lojasiewicz exponents are fundamentally valuative optimization problems. The central structural contribution of the paper is a finite-max principle. Under verifiable algebraic hypotheses, the a priori infinite valuative supremum bounding the Lojasiewicz exponent reduces to a finite maximum, and computes the Lojasiewicz exponent precisely. We identify two complementary mechanisms leading to this phenomenon: finite testing arising from normalized blowups and Noetherian Rees algebras, and attainment via compactness of normalized valuation spaces under linear boundedness assumptions. This finite-max framework yields strong structural consequences. We prove rigidity results showing that common extremal valuations force equality of Lojasiewicz ratios, and we establish stratification and stability phenomena for Lojasiewicz exponents in families, including fractional linearity and wall-chamber behavior along natural one-parameter deformations. The theory recovers and explains classical results in toric and Newton-polyhedral settings, particularly, for Newton nondegenete case, where the Lojasiewicz exponent is computed by finitely many toric divisorial valuations corresponding to facet data. Finally, we illustrate why the hypotheses underlying the finite-max principle are essential, delineating the precise scope of the theory.

math.AC↗

arXiv2026

Rational symbolic powers of ideals

We introduce and study rational symbolic powers of ideals in Noetherian rings. We give membership criteria for rational symbolic powers and discuss settings where they agree with integer symbolic powers. We investigate the binomial expansion formula for rational symbolic powers of mixed sums of ideals. Finally, we study rational symbolic powers of monomial ideals. In this case, we give a convex-geometric description of the rational symbolic powers. We also show that the filtration of rational symbolic powers of a monomial ideal is asymptotically stable and, as a consequence, deduce that the asymptotic regularity and asymptotic depth for this filtration exist.

math.AC↗

arXiv2026

Relative Hochster--Takayama formula and Cohen--Macaulay monomial ideal quotients

Hochster's and Takayama's formulas describes the multigraded components of local cohomology modules of monomial ideals in terms of simplicial complexes. In this paper, we develop a relative version of these formulas for quotients $I/J$ of monomial ideals, expressing the multigraded pieces of local cohomology modules of $I/J$ as reduced relative (co)homology of pairs of degree complexes. As an application, we obtain a relative Reisner criterion characterizing Cohen-Macaulay monomial ideal quotients. We further apply this relative Hochster--Takayama framework to modules arising from symbolic power filtrations, including symbolic quotients $I^{(t)}/I^{(t+1)}$ and symbolic-ordinary discrepancy module $I^{(t)}/I^t$. In particular, for a squarefree monomial ideal $I$, we give a precise classification of when $I^{(t)}/I^{(t+1)}$ is Cohen-Macaulay for all or, equivalently, for some $t \ge 2$. When $I$ is the edge ideal of a graph, we characterize the Cohen-Macaulayness of $I^{(t)}/I^t$ for all or, equivalently, for some sufficiently large $t$, and analyze the behavior of its dimension function.

math.AC↗

arXiv2025

Defect Functions Between Filtrations of Ideals

We introduce and study the defect function associated to a pair of filtrations of ideals, which generalizes the symbolic defect of ideals. Under the assumption that the Rees algebra of one filtration is Noetherian and that a natural graded module measuring the interaction between the filtrations is finitely generated over it, we show that the corresponding defect function is asymptotically a quasi-polynomial. Moreover, the defect function becomes eventually polynomial when the Rees algebra of the first filtration is standard graded. For filtrations arising from saturations and ordinary powers of monomial ideals, we further analyze the structure of the quasi-polynomial. We prove that the top two coefficients of the eventual quasi-polynomial are constant under natural hypotheses.

math.AC↗

arXiv2025

Analytic spread of binomial edge ideals

We investigate the analytic spread of binomial edge ideals of finite simple graphs. We provide tight bounds for this invariant in general. For special families of graphs (e.g., closed graphs, pseudo-forests), we compute the exact value for the analytic spread of the corresponding binomial edge ideals via combinatorial and convex geometric means.

math.AC↗

arXiv2025

Asymptotic regularity of graded families of ideals

We show that the asymptotic regularity of a graded family $(I_n)_{n \ge 0}$ of homogeneous ideals in a reduced standard graded algebra, i.e., the limit $\lim_{n \rightarrow \infty} \text{reg } I_n/n$, exists in several cases; for example, when the family $(I_n)_{n \ge 0}$ consists of artinian ideals, or Cohen-Macaulay ideals of the same codimension over an uncountable base field of characteristic $0$, or when its Rees algebra is Noetherian. Many applications, including simplifications and generalizations of previously known results on symbolic powers and integral closures of powers of homogeneous ideals, are discussed. We provide a combinatorial interpretation of the limit $\lim_{n \rightarrow \infty} \text{reg } I_n/n$ in terms of the associated Newton--Okounkov region in various situations. We give a negative answer to the question of whether the limits $\lim_{n \rightarrow \infty} \text{reg } (I_1^n + \dots + I_p^n)/n$ and $\lim_{n \rightarrow \infty} \text{reg } (I_1^n \cap \cdots \cap I_p^n)/n$ exist, for $p \ge 2$ and homogeneous ideals $I_1, \dots, I_p$. We also examine ample evidence supporting a negative answer to the question of whether the asymptotic regularity of the family of symbolic powers of a homogeneous ideal always exists. Our work presents explicit Gröbner basis construction for ideals of the form $Q^n + (f^k)$, where $Q$ is a monomial ideal, $f$ is a polynomial in the polynomial ring in 4 variables over a field of characteristic $2$.

math.AC↗

arXiv2024

Resurgence number and convex body associated to pairs of graded families of ideals

We discuss how to understand the asymptotic resurgence number of a pair of graded families of ideals from combinatorial data of their associated convex bodies. When the families consist of monomial ideals, the convex bodies being considered are the Newton-Okounkov bodies of the families. When ideals in the second family are classical invariant ideals, for instance, determinantal ideals or ideals of Pfaffians, these convex bodies are constructed from the associated Rees packages.

math.AC↗