Fixed-degree kernels over a number field

Determine which rational polygons P have a nonempty K-coefficient arithmetic kernel Ker_d(P,K) for a fixed number field K and degree bound d; determine whether the number of vertices or reflex vertices can bound a sufficient degree, and compare this kernel with the set of rational guards that see every real point of P along real-coefficient carriers of degree at most d.

Background

The arithmetic kernel consists of rational guards that can see every target using carriers of bounded degree over a specified coefficient field K, with no coefficient-height bound. At degree one, the K-coefficient kernel coincides with the classical kernel intersected with the rational points, whereas quadratic carriers can enlarge it, as demonstrated by the L-shaped gallery example.

The paper also shows that rational carriers may fail to provide a whole-gallery guard at a fixed degree, even in convex polygons, because algebraic targets can have relation degree exceeding the permitted carrier degree. The problem seeks structural criteria for nonemptiness and quantitative bounds in terms of polygonal complexity, together with a comparison to the analogous real-coefficient visibility notion.