The Zero Locus of the $F$-triangle

TL;DR

This paper investigates the zero locus of Chapoton's $F$-triangle polynomials, demonstrating that for low ranks, the roots are real, distinct, and vary monotonically with respect to a parameter, providing new insights into related polynomial families.

Contribution

The paper introduces a generalized concept of $F$-triangles and confirms the expected root behavior for low ranks, proposing conjectures for higher ranks.

Findings

For low rank, roots are real and distinct in [0,1].

Roots vary monotonically with respect to parameter y.

Provides new insights into zero loci of related polynomials.

Abstract

We are interested in the zero locus of a Chapoton's $F$-triangle as a polynomial in two real variables $x$ and $y$. An expectation is that (1) the $F$-triangle of rank $l$ as a polynomial in $x$ for each fixed $y\in[0,1]$, has exactly $l$ distinct real roots in $[0,1]$, and (2) $i$-th root $x_i(y)$ ($1 \le i \le l$) as a function on $y \in [0,1]$ is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of $F$-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above expectation is true. We formulate inductive conjectures and questions for further rank cases. This study gives a new insight on the zero loci of $f^+$- and $f$-polynomials.