arXiv subjects

Naser Talebizadeh Sardari

Publications and source records attributed to Naser Talebizadeh Sardari.

6 recordsLinked to original sources

arXiv2026

CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of \textbf{756 images} of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only \textbf{71.1\%} tree-generation accuracy on CurveBench-Easy and \textbf{19.1\%} on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over \texttt{Qwen-3-VL-8B-Thinking} from \textbf{2.8\%} to \textbf{33.3\%} tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.

cs.CV↗

arXiv2024

New Lower Bounds for the Schur-Siegel-Smyth Trace Problem

We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known bound to 1.80203. Our method adds new constraints to Smyth's linear programming method to decrease the number of variables required in the new problem of interest. This allows for faster convergence recovering Schur's bound in the simplest case and Siegel's bound in the second simplest case of our new family of bounds. We also prove the existence of a unique optimal solution to our newly phrased problem and express the optimal solution in terms of polynomials. Lastly, we solve this new problem numerically with a gradient descent algorithm to attain the new bound 1.80203.

math.NT↗

arXiv2024

Limiting distributions of conjugate algebraic integers

Let $Σ\subset \mathbb{C}$ be a compact subset of the complex plane, and $μ$ be a probability distribution on $Σ$. We give necessary and sufficient conditions for $μ$ to be the weak* limit of a sequence of uniform probability measures on a complete set of conjugate algebraic integers lying eventually in any open set containing $Σ$. Given $n\geq 0$, any probability measure $μ$ satisfying our necessary conditions, and any open set $D$ containing $Σ$, we develop and implement a polynomial time algorithm in $n$ that returns an integral monic irreducible polynomial of degree $n$ such that all of its roots are inside $D$ and their root distributions converge weakly to $μ$ as $n\to \infty$. We also prove our theorem for $Σ\subset \mathbb{R}$ and open sets inside $\mathbb{R}$ that recovers Smith's main theorem \cite{Smith} as special case. Given any finite field $\mathbb{F}_q$ and any integer $n$, our algorithm returns infinitely many abelian varieties over $\mathbb{F}_q$ which are not isogenous to the Jacobian of any curve over $\mathbb{F}_{q^n}$.

math.NT↗

arXiv2024

A quantitative converse of Fekete's theorem

Given a compact subset $Σ\subset \mathbb{R}$ (or $\mathbb{C}$) with logarithmic capacity greater than zero, we construct an explicit family of probability measures supported on $Σ$ such that their closure is all the possible weak limit measures of complete sets of conjugate algebraic integers lying inside $Σ$. We give an asymptotic formula for the number of algebraic integers with given degree and prescribed distribution. We exploit the algorithmic nature of our approach to give a family of upper bounds that converges to the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the best previously known upper bound on the Schur-Siegel-Smyth trace problem to 1.8216.

math.NT↗

arXiv2021

Higher Fourier interpolation on the plane

Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $μ(τ)dτ$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where $\Im(τ),\Im(-\frac{1}τ)>\sin\left(\fracπ{l}\right).$ Let $f(x)=\int e^{iπτ|x|^2}dμ(τ)$ and $\mathcal{F}(f)$ be its Fourier transform, where $x\in \R^2.$ For every integer $k\geq 0$ and $x\in \R^2,$ we express $f(x)$ in terms of the values of $\frac{d^k f}{du^k}$ and $\frac{d^k \mathcal{F}(f)}{du^k}$ at $u=\frac{2n}λ,$ where $n$ is a non-negative integer, $u=|x|^2$ and $λ=2\cos\left(\fracπ{l}\right).$ We show that the condition $\Im(τ),\Im(-\frac{1}τ)>\sin\left(\fracπ{l}\right)$ is optimal. We also identify the summation formulas among the values of $\frac{d^k f}{du^k}$ and $\frac{d^k \mathcal{F}(f)}{du^k}$ at $u=\frac{2n}λ,$ with the space of holomorphic modular forms of weight $2k+1$ of the Hecke triangle group $(2,l,\infty)$. Using our formulas for $l=6$ and developing new methods, we prove a conjecture of Cohn, Kumar, Miller, Radchenko and Viazovska~\cite[Conjecture 7.5]{Maryna3}. This conjecture was motivated by the universal optimality of the hexagonal lattice.

math.NT↗

arXiv2021

Intersecting geodesics on the modular surface

We introduce the \textit{modular intersection kernel}, and we use it to study how geodesics intersect on the full modular surface $\mathbb{X}=PSL_2\left(\mathbb{Z}\right) \backslash \mathbb{H}$. Let $C_d$ be the union of closed geodesics with discriminant $d$ and let $β\subset \mathbb{X}$ be a compact geodesic segment. As an application of Duke's theorem to the modular intersection kernel, we prove that $ \{\left(p,θ_p\right)~:~p\in β\cap C_d\}$ becomes equidistributed with respect to $\sin θds dθ$ on $β\times [0,π]$ with a power saving rate as $d \to +\infty$. Here $θ_p$ is the angle of intersection between $β$ and $C_d$ at $p$. This settles the main conjectures introduced by Rickards \cite{rick}. We prove a similar result for the distribution of angles of intersections between $C_{d_1}$ and $C_{d_2}$ with a power-saving rate in $d_1$ and $d_2$ as $d_1+d_2 \to \infty$. Previous works on the corresponding problem for compact surfaces do not apply to $\mathbb{X}$, because of the singular behavior of the modular intersection kernel near the cusp. We analyze the singular behavior of the modular intersection kernel by approximating it by general (not necessarily spherical) point-pair invariants on $PSL_2\left(\mathbb{Z}\right) \backslash PSL_2\left(\mathbb{R}\right)$ and then by studying their full spectral expansion.

math.NT↗