Sharper effective degree and height bounds

Sharpen the effective degree and coefficient-height bounds in the arithmetic arc-lifting theorem by expressing them in terms of polygon vertex data, endpoint degrees and heights, and intrinsic arc geometry, including boundary distance from an endpoint and the inward tangent angle; determine the possible degree–height tradeoffs.

Background

The effective arithmetic arc-lifting theorem constructs a rational, absolutely irreducible carrier joining a rational interior guard to an algebraic target while keeping a regular sight arc inside a rational polygon. Its explicit bounds are computable but depend on coarse interpolation, approximation, and cutoff constants.

The paper notes that direct geometric constructions, such as conics and the triangle-based construction, can yield substantially smaller budgets. The open problem is to replace the coarse general estimates with sharper bounds reflecting the actual polygon, endpoint arithmetic, and geometry of the required arc.