arXiv subjects

Venkata Siddharth Pendyala

Publications and source records attributed to Venkata Siddharth Pendyala.

6 recordsLinked to original sources

arXiv2026

Thirteen Results in Magnitude Theory

We prove thirteen results concerning three aspects of magnitude theory. The first four concern small-scale magnitude and microscopic weightings. We identify the derivative of the magnitude function at the origin with maximal energy, allowing the value $+\infty$, and construct finite spaces for which small-scale magnitude diverges or for which the one-point property and finite concentration do not produce a microscopic weighting. A fifth result disproves the conjectured monotonicity of magnitude by giving a fourteen-point metric space of strict negative type whose magnitude function decreases. The remaining eight results concern continuity and dimension. Magnitude is Hausdorff-continuous at finite subsets of finite-dimensional positive definite normed spaces, but need not be continuous on compact subsets, even under convexity, geodesicity, bounded tractability, or simultaneous Hausdorff and Lipschitz convergence. The same constructions disprove an $\ell_1$-intrinsic-volume formula and two tractability conjectures. Finally, we construct a countable compact space of negative type whose Minkowski dimension is zero and whose lower and upper magnitude dimensions are both infinite.

math.MG↗

arXiv2026

Fefferman-Szegő kernels and finite-type rigidity on egg domains

We compute the Fefferman boundary measure and the associated Fefferman--Szegő kernel for the egg domains $$ E_{n,m}=\{(z,w)\in\mathbb{C}^{n-1}\times\mathbb{C}:\ |z|^2+|w|^{2m}<1\}. $$ The kernel is given both by an orthogonal monomial expansion and by a closed form in a natural auxiliary finite-type variable; its diagonal weak-boundary exponent recovers the integer $m$. For $n\ge2$, the associated Fefferman--Szegő metric has constant scalar curvature only in the ball case $m=1$, and the Kähler--Einstein, constant Ricci-spectrum, and Bergman-proportionality statements follow as corollaries of the same calculation.

math.CV↗

arXiv2026

A Degree-Four Lemniscate Path Theorem

We prove the degree-four case of a path problem of Erdős, Herzog, and Piranian. If $f$ is monic of degree four and all zeros of $f$, counted with multiplicity, lie in the open unit disk, then two zeros from this list can be joined inside $$\{z:|f(z)|<1\}$$ by a possibly degenerate polygonal path of length less than $2$.

math.CV↗

arXiv2026

Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős

Let $f(z)=\prod_{j=1}^{n}(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\{z\in\mathbb{C}: |z|\leq 1,\ |f(z)|\leq 1\}$. Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\partial\mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\sqrt{\log n}\leq S(n)\leq πn$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erdős. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.

math.CV↗

arXiv2026

The Fefferman-Szegő Sphericity Criterion in Complex Dimension Three

We establish a Fefferman-Szegő characterization of local CR sphericity for smoothly bounded strongly pseudoconvex domains in complex dimension three. We derive the boundary expansion of the normalized determinant of the Fefferman-Szegő metric and prove that its second-order coefficient is a universal multiple of the squared Chern-Moser curvature. Hence, vanishing of the second-order deviation from the ball model is equivalent to local sphericity. A logarithmic stability theorem for the associated Monge-Ampère determinant controls the remainder and completes the dimension-three case.

math.CV↗

arXiv2026

Sharp order in Erdős's minimum-area problem for polynomial lemniscates

For a monic polynomial $p$, its filled unit lemniscate is the planar set ${z: |p(z)|<1}$. Let $κ_n(K,1)$ denote the least possible area of this set among monic polynomials of degree $n$ whose zeros lie in a compact set $K$. We prove that there are absolute constants $c,C>0$ such that $c/\log n \leq κ_n(\overline{\mathbb{D}},1) \leq κ_n(\mathbb{T},1) \leq C/\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. The upper bound is obtained by combining a quantitative Faber-polynomial separator for a thin keyhole domain with an equal-weight midpoint discretization that preserves the degree exactly. We also deduce that the critical boundary-zero minimizers form a normal family in $\mathbb{D}$.

math.CV↗