The quadratic difference conjecture for symplectic pairs
Let and be monic polynomials of degree , let be a vector space, and let be a symplectic pair on . A symplectic -difference is a symplectic pair satisfying the -difference condition, while a -difference in is an endomorphism satisfying the corresponding algebraic condition. Quadratic difference conjecture. For a symplectic pair to be a symplectic -difference, it is necessary and sufficient that be a -difference in . This would extend the previously established equivalence for to arbitrary pairs of monic quadratic polynomials; the source gives no resolution, so the conjecture remains open.
References
Primary source
Clément de Seguins Pazzis, “The quadratic sum problem for symplectic pairs”, arXiv:2305.19340 (2023).
Progress summary
A 2023 paper gives an explicit counterexample, and a reader points to it, so the conjecture is claimed false but has not been independently verified here.
Clément de Seguins Pazzis’s 2023 paper treats the proposed equivalence as Conjecture 1. It claims that the equivalence fails for arbitrary monic quadratic polynomials.
2023 counterexample
For irreducible , a symplectic space of dimension with odd, and the pair , the endomorphism is a -difference, but is not a symplectic -difference. The same source records a surviving equivalence when at least one of or has a root in the base field.
Posted attempt
A reader identifies page 6 of the paper as a counterexample; this is an unverified reader-written attempt rather than independent mathematical confirmation.
Current status (as of August 2026): The universal conjecture is claimed refuted by the 2023 counterexample, while restricted cases remain valid and no independent verification of the refutation was found.
Sources
Solutions 1
CounterexampleThis solution needs a summarySee full solution
The paper were this is taken from actually gives a counterexample: see page 6 of https://arxiv.org/pdf/2305.19340