Let g∈widehatmathbfP, and define F_m(x,y) by fracd^mdx^mf(g(x))=∑_k f^(k)(g(x))A_m,k(x),qquad F_m(x,y)=∑_k y^kA_m,k(x). The Faà di Bruno real-rootedness conjecture. For every alpha∈R and m=0,1,…, F_m(alpha,y) has only real roots and F_m+1(alpha,y)lessdot F_m(alpha,y). The source says this is known in the first few cases, for e^x, and for x^d, but leaves the general assertion open.

The Faà di Bruno real-rootedness conjecture

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Let $g\in\widehat{\mathbf{P}}$ , and define $F_m(x,y)$ by

\frac{d^m}{dx^m}f(g(x))=\sum_k f^{(k)}(g(x))A_{m,k}(x),\qquad F_m(x,y)=\sum_k y^kA_{m,k}(x).

The Faà di Bruno real-rootedness conjecture. For every $\alpha\in\mathbb{R}$ and $m=0,1,\dots$ , $F_m(\alpha,y)$ has only real roots and

F_{m+1}(\alpha,y)\lessdot F_m(\alpha,y).

The source says this is known in the first few cases, for $e^x$ , and for $x^d$ , but leaves the general assertion open.

Progress summary

Open

No public discussion or published progress on this conjecture was found.

No public discussion or published progress on the Faà di Bruno real-rootedness conjecture was found in the retrieved sources.

Current status (as of August 2026): The conjecture appears open, with no recorded public activity or verified progress.

Sources & referencesView supporting material

Primary source

Steve Fisk, “Polynomials, roots, and interlacing”, arXiv:math/0612833 (2008).

1 week ago

The conjecture is false already for $m=4$ , the first case beyond those established in the source.

Take

g(x)=(x-1)(x+2)^3,\qquad \alpha=0.

All zeros of $g$ are real, so $g\in\mathcal P\subseteq\widehat{\mathcal P}$ , exactly as required. Direct differentiation gives

g'(0)=-4,\qquad g''(0)=12,\qquad g'''(0)=30,\qquad g^{(4)}(0)=24.

The Faà di Bruno polynomial at order four is

F_4(\alpha,y) =(g'(\alpha))^4y^4 +6g''(\alpha)(g'(\alpha))^2y^3 +\bigl(4g'''(\alpha)g'(\alpha)+3(g''(\alpha))^2\bigr)y^2 +g^{(4)}(\alpha)y.

Consequently,

F_4(0,y)=8y\bigl(32y^3+144y^2-6y+3\bigr).

For

q(y)=32y^3+144y^2-6y+3

the discriminant is

\operatorname{disc}(q) =-36{,}799{,}488 =-2^{10}\cdot3^3\cdot11^3<0.

A real cubic with negative discriminant has one real zero and a nonreal complex-conjugate pair. Therefore $F_4(0,y)$ is not real-rooted. This contradicts the conjectured real-rootedness and, consequently, also precludes the proposed universal consecutive interlacing.

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Shivam Patel · 1 day ago