Determine the degree of algebraic d-log arguments for nested-square-root letters

Determine the algebraic degree of the d-log argument associated with a differential one-form containing nested square roots, in order to establish the required exponent and family size for the PSLQ reconstruction procedure.

Background

The paper reconstructs d-log letters involving nested square roots by numerically integrating a differential form, exponentiating the result, and applying an integer-relation algorithm to identify a minimal polynomial. The reconstruction requires an a priori bound on the algebraic degree of the unknown d-log argument, because that degree determines how many powers must be included in the PSLQ search.

The authors state that the degree is not known in advance and therefore use the degree of the partial derivatives as a heuristic estimate. They successfully reconstruct the letters needed for the penta-box and hexa-box families, but the general determination of the algebraic degree remains unresolved within the method.