Algebraic structure underlying pole-skipping points

The holographic Green’s function becomes ambiguous, taking the indeterminate form “ <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>0</a:mn> <a:mo>/</a:mo> <a:mn>0</a:mn> </a:math> ,” at an infinite set of special frequencies and momenta known as “pole-skipping points.” In this work, we propose that these pole-skipping points can be used to reconstruct both the interior and exterior geometry of a static, planar-symmetric black hole in the bulk. The entire reconstruction procedure is fully analytical and only involves solving a system of linear equations. We demonstrate its effectiveness across various backgrounds, including the Bañados-Teitelboim-Zanelli black hole, its <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>T</c:mi> <c:mover accent="true"> <c:mi>T</c:mi> <c:mo stretchy="false">¯</c:mo> </c:mover> </c:math> -deformed counterparts, as well as geometries with Lifshitz scaling and hyperscaling-violation. Within this framework, other geometric quantities, such as the vacuum Einstein equations, can also be reinterpreted directly in terms of pole-skipping data. Moreover, our approach reveals a hidden algebraic structure governing the pole-skipping points of Klein-Gordon equations of the form <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo stretchy="false">(</g:mo> <g:msup> <g:mo>∇</g:mo> <g:mn>2</g:mn> </g:msup> <g:mo>+</g:mo> <g:mi>V</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>r</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo stretchy="false">)</g:mo> <g:mi>ϕ</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>r</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo>=</g:mo> <g:mn>0</g:mn> </g:math> : only a subset of these points is independent, while the remainder is constrained by an equal number of homogeneous polynomial identities in the pole-skipping momenta. These identities are universal, as confirmed by their validity across a broad class of bulk geometries with varying dimensionality, boundary asymptotics, and perturbation modes.