## 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*∈[0,1]*has exactly* *l* *distinct real roots in* [0,1], *and* (2) *ith* *root* *x* _{ i }(*y*) (1≤*i*≤*l*) *as a function on* *y*∈[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.

## Keywords

*F*-triangle

*A*-triangle Polyhedral cone Real zero loci

## Mathematics Subject Classification (2010)

05A99## Notes

### Acknowledgments

The author express his gratitudes to Christos Athanasiadis, Frederic Chapoton, Christian Krattenthaler, and Tadashi Ishibe for discussions and interests, and to Yoshihisa Obayashi for the helps in drawing figures. A particular gratitude goes to Frederic Chapoton, who, after looking at an early version of the present note, informed the author other examples of *F*-triangles which supported the author to formulate conjecture in terms of *A*-polynomials.

This research was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, by JSPS KAKENHI Grant Number 25247004, and by JSPS bilateral Japan - Russia Research Cooperative Program.

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