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Zhengyu Tao

Publications and source records attributed to Zhengyu Tao.

9 recordsLinked to original sources
arXiv2026

The weak Chinburg conjecture on Mahler measures

For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(χ_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $χ_{-f}$ is the quadratic Dirichlet character associated with $-f$. This proves the weak Chinburg conjecture. An independent construction using Bloch cycles yields a stronger result in two variables: there exists a nonzero polynomial $R_f\in\mathbb{Q}[x,y]$ satisfying \[ m(R_f)=4wL'(χ_{-f},-1), \] where $w$ is the number of roots of unity in $\mathbb{Q}(\sqrt{-f})$.

math.NT↗
arXiv2026

ForeTac-VLA: A Forecasting-Based Tactile-Vision-Language-Action Model for Contact-Rich Robotic Manipulation

Vision-language-action (VLA) models have demonstrated strong capabilities in robotic manipulation, yet their reliance on visual perception limits robustness in contact-rich environments, where critical physical interaction states may not be visually observable. Existing tactile-enhanced VLA methods improve physical grounding using observed tactile feedback, but most remain largely reactive rather than explicitly modeling how contact may evolve. Therefore, we propose ForeTac-VLA, a forecasting-based tactile-vision-language fusion model that predicts future tactile states to guide action generation. Specifically, ForeTac-VLA encodes recent tactile observations into temporal representations and integrates them with vision-language features through bidirectional cross-attention. Further, a transformer-based forecasting module predicts multi-step future tactile states, enabling the model to reason jointly over observed and anticipated contact. Finally, the fused multimodal representations and predicted future tactile states are fed into the VLA backbone to condition action generation. To stabilize training, a ground-truth-to-prediction curriculum is employed when early forecasts are unreliable. Across four real-world contact-rich manipulation tasks, ForeTac-VLA achieves an average success rate of 95%, outperforming the fine-tuned VLA model by 36.25 percentage points and state-of-the-art tactile-enhanced VLA baselines by over 22 percentage points. ForeTac-VLA also maintains strong performance under low-illumination and visually cluttered conditions. Video demonstrations can be found on https://foretac-vla.github.io/

cs.RO↗
arXiv2026

Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups

For an odd prime $p$, let $A_j$ be the $ω^j$-eigenspace of the $p$-primary class group of $\mathbb Q(ζ_p)$. Fix an even integer $d\ge4$, put $N=(p-1)/d$, and let $U_d=(\mathbb Z/d\mathbb Z)^\times$. For a relative density-one set of primes $p\equiv d+1\pmod{2d}$, we prove that the odd block $\bigoplus_{a\in U_d}A_{aN}$ has order at most $p^{φ(d)/2-1}$, and reflection shows that the even block $\bigoplus_{a\in U_d}A_{p-aN}$ has $p$-rank at most $φ(d)/2-1$. For each $d\in\{4,6\}$, both blocks vanish for a relative density-one set of primes $p\equiv d+1\pmod{2d}$; in particular, the even components vanish in accordance with Vandiver's conjecture. For each $d\in\{8,10,12\}$, the odd block has order at most $p$ for a relative density-one set of primes in the same progression, so every summand $A_{aN}$, $a\in U_d$, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet $L$-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves $A_{34}=0$ for every odd prime $p$; a single-file PARI/GP program reproduces the calculation.

math.NT↗
arXiv2026

Determinants of Mahler measures and special values of $L$-functions

We consider Mahler measures of two well-studied families of bivariate polynomials, namely $P_t=x+x^{-1}+y+y^{-1}+\sqrt{t}$ and $Q_t=x^3+y^3+1-\sqrt[3]{t}xy$, where $t$ is a complex parameter. In the cases when the zero loci of these polynomials define CM elliptic curves over number fields, we derive general formulas for their Mahler measures in terms of $L$-values of cusp forms. For each family, we also classify all possible values of $t$ in number fields of degree not exceeding $4$ for which the corresponding elliptic curves have complex multiplication. Finally, for all such values of $t$ in totally real number fields of degree $n=2$ and $n=4$, corresponding to elliptic curves $\mathcal{F}_t$ (resp. $\mathcal{C}_t$), we prove that determinants of $n\times n$ matrices whose entries are Mahler measures corresponding to their Galois conjugates are non-zero rational multiples of $L^{(n)}(\mathcal{F}_t,0)$ (resp. $L^{(n)}(\mathcal{C}_t,0)$).

math.NT↗
arXiv2026

Counterexamples to Stanley's conjecture on dimer coverings

Let $Q_k(x)$ be Stanley's explicit denominator for the dimer-covering generating function $F_k(x)=\sum_{n\ge0}A_{k,n}x^n$ of $k\times n$ rectangles. Stanley conjectured in 1985 that $Q_k(x)$ has only simple roots; this longstanding conjecture was recently recorded in Lai's list of open problems on tilings (see [6, Problem 33]). We disprove the conjecture by proving that $Q_{14h-1}(x)$ and $Q_{30h-1}(x)$ have repeated roots for every $h\ge1$; in particular, $k=13$ is the smallest counterexample. The construction comes from two exceptional multiplicative identities among trigonometric algebraic units. We further propose a conjecture concerning this class of trigonometric identities, which appears to be related to Robinson's problem on primitive Pell factors.

math.CO↗
arXiv2024

CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$

We study the Mahler measures of the polynomial family $Q_k(x,y) = x^3+y^3+1-kxy$ using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers $\leqslant 3$, we employ these points to derive interesting formulas that link the Mahler measures of $Q_k(x,y)$ to $L$-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure $\tilde{n}(k)$ introduced by Samart recently. For $k=\sqrt[3]{729\pm405\sqrt{3}}$, we also prove an equality that expresses a $2\times 2$ determinant with entries the Mahler measures of $Q_k(x,y)$ as some multiple of the $L$-value of two isogenous elliptic curves over $\mathbb{Q}(\sqrt{3})$.

math.NT↗
arXiv2024

Mahler measures and $L$-values of elliptic curves over real quadratic fields

A famous formula of Rodriguez Villegas shows that the Mahler measures $m(k)$ of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series. We prove that the degree of $k$ in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive $28$ new identities linking $m(k)$ to $L$-values of cusp forms. Guided by Beilinson's conjecture, we also prove $5$ formulas that express $L$-values of CM elliptic curves over real quadratic fields to some $2\times 2$ determinants of $m(k)$. This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when $k=4\pm 4\sqrt{2}$.

math.NT↗
arXiv2022

Some new Ramanujan-Sato series for $1/π$

We derive 10 new Ramanujan-Sato series of $1/π$ by using the method of Huber, Schultz and Ye. The levels of these series are 14, 15, 16, 20, 21, 22, 26, 35, 39.

math.NT↗