Problems Around Polynomials — Conjecture 8

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\geq1$ and $f(z)=\sum_{j=0}^n a_jz^j$ with $a_j>0$ for every $j$. Define the weighted tropical polynomial on $\mathbb R$ by \[T_f(t)=\max_{0\leq j\leq n}\left(\log a_j+jt+\log\binom nj\right).\] Its corners are the distinct points where this continuous piecewise-linear function is not differentiable. Let $v_T(f)$ be the number of corners and $N_{\mathbb R}(f)$ the number of real roots, counted with multiplicity.

Statement recorded in the supplied source.

Does every such positive-coefficient polynomial satisfy \[N_{\mathbb R}(f)\leq v_T(f)?\] [S2]

Short English statement

Does the number of real roots never exceed the number of corners of the binomially weighted tropical coefficient graph?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2200013 (AMR-021-0013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boris Shapiro, Problems Around Polynomials -- The Good, The Bad and The Ugly, Arnold Mathematical Journal 1 (2015), pp.91--104, Conjecture 8 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1503.05295.