Discriminants of derivatives: square graphs and sums of squares
Abstract
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≤ n≤22, but it is a sum of squares for every n≥4. 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≥ 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≥4, and we conjecture that such a factorization exists for every derivative; this is verified in the centered outlier limit through terminal degree six.
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BibTeX
@misc{riener2026discriminants,
title = {{Discriminants of derivatives: square graphs and sums of squares}},
author = {Cordian Riener and Boris Shapiro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.25743},
eprint = {2605.25743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25743},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discriminants of derivatives: square graphs and sums of squares
AU - Cordian Riener
AU - Boris Shapiro
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2605.25743
UR - https://arxiv.org/abs/2605.25743
AB - 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≤ n≤22, but it is a sum of squares for every n≥4. 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≥ 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≥4, and we conjecture that such a factorization exists for every derivative; this is verified in the centered outlier limit through terminal degree six.
ER -