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Thanh Vu

Publications and source records attributed to Thanh Vu.

At least 19 recordsLinked to original sources
arXiv2026

v-numbers of symbolic-ordinary discrepancy modules

In this paper, we study the ${\rm v}$-number of the symbolic-ordinary discrepancy modules $M_t(I) = I^{(t)} / I^t$ over standard graded Noetherian rings. We show that this coincides with the ${\rm v}$-number of $I^t$ at embedded primes. Moreover, under mild assumptions on $I$, we show that the ${\rm v}$-number of $N_t(I) = I^{(t)} / I^{(t+1)}$ is the same as the ${\rm v}$-number of $I^{(t+1)}$. Furthermore, we provide combinatorial formulas for these ${\rm v}$-numbers when $t = 2, 3$ and $I$ is the edge ideal of a graph. Consequently, we classify all graphs for which ${\rm v}(I^2) = {\rm v}(I^{(2)})$.

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

V-numbers of powers of cover ideals of unimodular hypergraphs

Let $H$ be a unimodular hypergraph with cover ideal $J(H)$. We prove that the local $v$-numbers of $J(H)^t$ are linear in $t$ for all $t\ge1$. We further show that the global $v$-number of $J(H)^t$ is linear in $t$ for all $t\ge n-1$. Finally, we prove that the global $v$-number of the powers of the cover ideal of any tree is linear in $t$ for all $t\ge1$.

math.AC↗
arXiv2026

V-numbers of symbolic powers of cover ideals of graphs

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ with $|V(G)|$ variables. We prove that the local $v$-number of the symbolic powers $J(G)^{(t)}$ is linear for all $t \ge 1$ when $G$ is bipartite, and quasi-linear with period two for all $t \ge 2$ when $G$ is non-bipartite. Furthermore, we provide explicit formulas for these invariants in terms of the combinatorial data of $G$.

math.AC↗
arXiv2026

Ordered alternating paths and the depth of symbolic powers of cover ideals of graphs

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ in $|V(G)|$ variables. For a matching $M$ of $G$, we denote by $\ell(M)$ the length of the longest $M$-alternating path in $G$. We define $α_t(G)$ to be the maximum size of an ordered matching $M$ of $G$ such that $\ell(M) \le 2t-1$. We then prove that $$\operatorname{depth}(S/J(G)^{(t)}) \le |V(G)| - 1 - α_t(G)$$ for all $t \ge 1$, where $J(G)^{(t)}$ denotes the $t$-th symbolic power of $J(G)$, and that equality holds when $G$ is a forest.

math.AC↗
arXiv2026

Projective dimension of powers of cover ideal of Ferrers graphs

Let $λ= (λ_1, \ldots, λ_n)$ be a partition with $λ_1 = m$. Denote by $J_λ$ the cover ideal in the polynomial ring \( S = k[x_1, \ldots, x_n, y_1, \ldots, y_m] \) associated to the Ferrers graph corresponding to $λ$. Let $d(λ)$ denote the number of distinct parts of $λ$. We prove that \[ \operatorname{pd}(S/J_λ^t) = \min\{t,\; d(λ)\} + 1 \] for all $t \ge 1$.

math.AC↗
arXiv2026

Enhancing LLM Medical Coding with Structured External Knowledge

Accurate medical coding requires consulting authoritative resources such as the ICD tabular list and coding guidelines. Existing LLM-based automated methods largely rely on LLMs' internal knowledge, which is prone to hallucination and cannot keep pace with guideline updates. We introduce RAG-Coding, an agentic, training-free method that augments LLMs with structured external knowledge: the tabular list is encoded as a knowledge graph capturing hierarchical and instructional code relationships, and the guidelines are distilled into concise, code-specific summaries rather than retrieved as raw text. To enable our study, we also introduce MDACE-2025, expert re-annotations of the MDACE dataset under the 2025 ICD-10-CM/PCS guidelines, adding code sequencing and justification comments. On MDACE, RAG-Coding outperforms the best LLM-based baseline by 3--13\% in micro-F1 across five LLM backbones, and achieves comparable micro- and macro-F1 to the supervised state-of-the-art, with higher recall ($+$11\%) at the cost of precision ($-$6\%). On MDACE-2025, RAG-Coding outperforms all baselines, demonstrating effective generalisation to updated guidelines. Ablations confirm stepwise gains, highlighting the importance of integrating structured external knowledge for LLM-based medical coding.

cs.CL↗
arXiv2026

Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs

Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \[ \depth\big(S/J(C_n)^{(t)}\big) = n - 1 - \left\lfloor \frac{tn}{2t+1} \right\rfloor \] for all $t \ge 2$ and $n \ge 3$, where $S = K[x_1,\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices.

math.AC↗
arXiv2026

Contractible independence complexes of trees

We show that the independence complex of a tree is contractible if and only if it can be reduced to a path \( P_n \) with \( n \equiv 1 \pmod{3} \) by a sequence of truncation moves at branching points. As a consequence of our method, we also characterize the trees for which the independence polynomial evaluated at \( -1 \) is equal to \( 1 \) or \( -1 \).

math.CO↗
arXiv2025

AgentEval: Generative Agents as Reliable Proxies for Human Evaluation of AI-Generated Content

Modern businesses are increasingly challenged by the time and expense required to generate and assess high-quality content. Human writers face time constraints, and extrinsic evaluations can be costly. While Large Language Models (LLMs) offer potential in content creation, concerns about the quality of AI-generated content persist. Traditional evaluation methods, like human surveys, further add operational costs, highlighting the need for efficient, automated solutions. This research introduces Generative Agents as a means to tackle these challenges. These agents can rapidly and cost-effectively evaluate AI-generated content, simulating human judgment by rating aspects such as coherence, interestingness, clarity, fairness, and relevance. By incorporating these agents, businesses can streamline content generation and ensure consistent, high-quality output while minimizing reliance on costly human evaluations. The study provides critical insights into enhancing LLMs for producing business-aligned, high-quality content, offering significant advancements in automated content generation and evaluation.

cs.AI↗
arXiv2025

MedDCR: Learning to Design Agentic Workflows for Medical Coding

Medical coding converts free-text clinical notes into standardized diagnostic and procedural codes, which are essential for billing, hospital operations, and medical research. Unlike ordinary text classification, it requires multi-step reasoning: extracting diagnostic concepts, applying guideline constraints, mapping to hierarchical codebooks, and ensuring cross-document consistency. Recent advances leverage agentic LLMs, but most rely on rigid, manually crafted workflows that fail to capture the nuance and variability of real-world documentation, leaving open the question of how to systematically learn effective workflows. We present MedDCR, a closed-loop framework that treats workflow design as a learning problem. A Designer proposes workflows, a Coder executes them, and a Reflector evaluates predictions and provides constructive feedback, while a memory archive preserves prior designs for reuse and iterative refinement. On benchmark datasets, MedDCR outperforms state-of-the-art baselines and produces interpretable, adaptable workflows that better reflect real coding practice, improving both the reliability and trustworthiness of automated systems.

cs.AI↗
arXiv2025

Mastering the Craft of Data Synthesis for CodeLLMs

Large language models (LLMs) have shown impressive performance in \emph{code} understanding and generation, making coding tasks a key focus for researchers due to their practical applications and value as a testbed for LLM evaluation. Data synthesis and filtering techniques have been widely adopted and shown to be highly effective in this context. In this paper, we present a focused survey and taxonomy of these techniques, emphasizing recent advancements. We highlight key challenges, explore future research directions, and offer practical guidance for new researchers entering the field.

cs.SE↗