6 papers
Finite-Defect Rigidity and the Minimum Spherical 4-Design on the Two-Sphere
Shalender Singh, Vishnupriya Singh
We prove that every equal-weight spherical -design on has at least twelve points. Since the regular icosahedron is a spherical -design, this determines the exa…
Repairing the refined-decoupling proof of the 5/4 planar pinned Falconer theorem
Shalender Singh, Vishnu Priya Singh
We show that the literal arbitrary-packet form of the refined-decoupling estimate printed in the Guth--Iosevich-Ou-Wang proof of the planar pinned Falconer theorem is false, even a…
From Liar Paradox to Incongruent Sets: A Normal Form for Self-Reference
Shalender Singh, Vishnu Priya Singh Parmar
We introduce incongruent normal form (INF), a structural representation for self-referential semantic sentences. An INF replaces a self-referential sentence with a finite family of…
An Algebraic Rigidity Framework for Order-Oblivious Deterministic Black-Box PIT of ROABPs
Shalender Singh, Vishnupriya Singh
Deterministic black-box polynomial identity testing (PIT) for read-once oblivious algebraic branching programs (ROABPs) is a central open problem in algebraic complexity, particula…
Inverse Falconer Distance Theorems over the Integer Residue Rings
Shalender Singh, Vishnupriya Singh
We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set…
Inverse Theorems for Point-Sphere Incidences over Finite Fields
Shalender Singh, Vishnu Priya Singh
Let Q be a nondegenerate quadratic form on , with q odd & . We prove a two-scale inverse theory for point-sphere incidences. At the maximal-incidence endpoint…