arXiv2026
The discriminants of the derivatives of a real-rooted polynomial are nonnegative polynomials in its roots, and we ask whether this nonnegativity can be certified algebraically in the root differences. We compare three such certificates: sums of squares, positive combinations of symmetrized monomials in squared root differences (the square-graph cone of Alexandersson and Shapiro), and sum-of-squares factorizations of the Hermite matrix of the derivative. We work in the terminal range, where the degree $r=n-k$ of the $k$-th derivative is fixed and the degree $n$ of the polynomial grows. For $r=3$ the discriminant lies in the square-graph cone for every $n$, and we prove the three-graph formula conjectured by Alexandersson and Shapiro. For $r=4$ the two cones separate: the discriminant belongs to the square-graph cone exactly for $4\le n\le22$, but it is a sum of squares for every $n\ge4$. The negative part is certified by an explicit linear functional, the positive part by a rational Gram certificate built from nine orbits of squares under the symmetric group. Thus the square-graph conjecture fails, already at $(n,k)=(23,19)$, while the sum-of-squares question has a positive answer for all $k\ge n-4$. Finally, we give explicit sum-of-squares factorizations of the Hermite matrices of the terminal cubic and quartic. The quartic factorization yields a uniform proof of the sum-of-squares property for all $n\ge4$, and we conjecture that such a factorization exists for every derivative; this is verified in the centered outlier limit through terminal degree six.