CERTAIN INVARIANT ALGEBRAIC SETS IN Sp × Sq

The paper’s main bounds for the number of invariant meridians and parallels on Sp,q (Theorem 3.3) match exactly the candidate’s formulas: at most C(p,2)(m1−1)+∑_{i=2}^{p+1} mi+1 meridians, and at most C(q,2)(mp+2−1)+∑_{j=1}^{q−1} mp+1+j parallels. The paper proves this via the extactic determinant with a “triangular” selection and degree-growth estimates for iterated Lie derivatives, relying on Proposition 2.2 (Darboux–extactic divisibility) to translate invariant hyperplanes into linear factors of the extactic polynomial . The candidate’s solution uses the same Darboux–extactic mechanism and the same triangular degree bookkeeping, differing mainly in a cosmetic restriction to the x- or y-block (setting y=0 or x=0) to simplify the degree growth. That restriction is not needed in the paper’s proof and, if taken literally, requires an extra finiteness assumption for the restricted fields to avoid the identically-zero extactic case; nonetheless, the candidate’s final bounds coincide with the paper and follow the same proof pattern. One minor presentational issue in the paper is a likely “least” vs “largest” typo in describing which extactic term controls the degree, but the intended triangular-degree argument is standard and yields the stated bounds .