Establish completeness of S'-units over curve-variable special primes

Establish the completeness of the S'-unit group over a finite set of places supported at a special prime of the curve variable, in order to produce the holomorphic-remainder certificate of non-elementarity for integrals such as A39, \(\int\arcsin(x\sqrt{1-x^2})\,dx\).

Background

The implementation of the parallel Risch–Norman method can produce a holomorphic-remainder certificate only when the relevant degree bounds and unit groups are known to be complete. For A39, the integrand ∫arcsin⁡(x1−x2) dx\int\arcsin(x\sqrt{1-x^2})\,dx leads to a genus-three curve with a special prime over the curve variable. The S'-units at that special prime are obtained only through a bounded norm search, so the implementation cannot verify that the search has found the complete unit group and therefore withholds the certificate.

The paper identifies determining the S'-unit group over a finite set of places of the curve as a problem on the Jacobian of the genus-three curve. Solving it, or automating the reduction from a primitive top generator to a curve integral, would convert A39's failed status into a certified non-elementarity result.

References

The one it does not return, $\int\arcsin(x\sqrt{1-x2})\,dx$, is not elementary --- the holomorphic-remainder certificate of Part II shows this after one integration by parts --- and in its own tower, a curve of genus three, the implementation withholds the certificate because the one hypothesis it cannot verify there is the completeness of the $S'$-units over a special prime of the curve variable.

— Parallel Integration over Simple Radical Extensions in Mixed Towers: Charlwood's Integrals  (2609.25616 - Blake, 22 Sep 2026) in Abstract; Section 'Assessment of the implementation', subsection 'Limitations', item (L1)