Discriminants of derivatives and symmetric difference polynomials

Abstract

Let $P$ be a monic polynomial of degree $n$ with roots $x_1,\ldots,x_n$. We study the discriminants of the derivatives $P^{(k)}$ as symmetric translation-invariant polynomials in the original roots. Alexandersson and Shapiro conjectured that every such discriminant belongs to the cone generated by symmetrized graph monomials with even edge multiplicities. We obtain a sharp positive/negative picture in the first terminal cases. For the terminal cubic family $k=n-3$ we prove the conjecture for every $n\ge3$, and for $n\ge5$ obtain the three-graph formula conjectured in \cite[Example~3]{AS}. For the terminal quartic family we show, by exact finite computation, that $\disc(P^{(n-4)})$ belongs to the square-graph cone if and only if $4\le n\le22$. Thus the general square-graph cone conjecture fails already at $(n,k)=(23,19)$ and fails for every $k=n-4$ with $n\ge23$. The negative result is certified by an explicit linear functional which is nonnegative on every degree-$12$ square-graph generator and strictly negative on the quartic discriminant. We also record central-moment formulas, the subset-average and finite Appell structure of normalized terminal polynomials, and the explicit quintic member.