Real-Rooted Integer Polynomials Satisfy a g-Theorem: Proof of a Conjecture of Brenti and Welker

In his abstract Conjectures on \(f\)-vectors in Oberwolfach Report 1/2008, V. Welker recorded the following conjecture of F. Brenti and himself (Conjecture 8 there). Let \(h(t)=1+h_1t+\dots+h_dt^d\) have positive integer coefficients and only real roots. Then \((h_0,\dots,h_d)\) satisfies a \(g\)-theorem: it increases strictly up to a first peak \(h_r\) and does not increase afterwards, and the differences \(g_i=h_i-h_{i-1}\), \(1\le i\le r\), satisfy Macaulay's condition \(g_{i+1}\le g_i^{\langle i\rangle}\). We prove this conjecture, with the classical pseudo-power \(g^{\langle i\rangle}\); the definition printed in the source contains a slip under which the source's own statement of the \(g\)-theorem would fail. No symmetry is assumed. More generally, the conclusion holds for every sequence of positive integers with \(h_0=1\) and \(ih_i^2\ge(i+1)h_{i-1}h_{i+1}\); by Newton's classical inequalities this class contains the coefficient sequences of real-rooted polynomials. The proof combines the elementary inequality \((k+1)!\,g_{k+1}\le(k!\,g_k)^{(k+1)/k}\) for the differences of such sequences with a lower bound for Macaulay's pseudo-power, \(a^{\langle k\rangle}\ge(k!\,a)^{(k+1)/k}/(k+1)!\) for all positive integers \(a\) and \(k\); both inequalities are asymptotically sharp. The palindromic case, in which the conclusion says that \(h\) is the \(h\)-vector of a simplicial polytope, was proved by Guo and Kang (2026, unrefereed preprint) by a different method; Theorem 9 of the source (Kubitzke and Nevo) and a theorem of Bell and Skandera also follow. The contribution of this note is the lower bound for the pseudo-power and the resulting proof of the general case. The stronger Kruskal–Katona version, asked by Kubitzke and Welker, remains open. The first conjecture of the corpus record, on real roots for barycentric subdivisions of non-simplicial polytopes, is not treated; as far as we know, it is open. This is an unrefereed note.