Second-kind representatives for irreducible elliptic curves

Develop a systematic treatment of second-kind differential representatives and corresponding rational basis candidates when the quartic defining the elliptic curve is irreducible over the rational function field of the kinematic variables.

Background

The construction uses a numerator for the second-kind differential that is rational when the quartic polynomial defining the elliptic curve factorizes into suitable quadratic factors. If the quartic is irreducible, the paper states that a different numerator built from fully symmetric combinations of the roots should be used.

A general, systematic construction for this irreducible case is not provided. Such a construction is needed to extend the elliptic-leading-singularity prescription beyond the factorization patterns present in the examples studied.

References

We leave a systematic treatment of this case for future work.

— How to choose a good rational basis for elliptic Feynman integrals?  (2609.08875 - Chaubey et al., 8 Sep 2026) in Footnote following Eq. (3.8), Section “Rational bases from elliptic leading singularities”

We can implement this requirement completely for families with a simple subsector hierarchy, such as the SR family, but we do not find a general integrand-level prescription for more complicated examples.

— How to choose a good rational basis for elliptic Feynman integrals?  (2609.08875 - Chaubey et al., 8 Sep 2026) in Section 4, subsection discussing the construction of J_R,phi, before Eq. (4.8)