Variations on the theme of classical discriminant

The classical discriminant $\Delta_n(f)$ of a degree $n$ polynomial $f(x)$ is an irreducible homogeneous polynomial of degree $2n-2$ on the coefficients $a_0, \ldots, a_n$ of $f$ that vanishes if and only if $f$ has a multiple zero. I will explain a tropical proof of the theorem of Gelfand, Kapranov and Zelevinsky (1990) that identifies the Newton polytope $P_n$ of $\Delta_n$ with an $(n-1)$-dimensional combinatorial cube obtained from the classical root system of type $A_{n-1}$. Recently Mikhalkin and Tsikh (2017) discovered a nice factorization property for truncations of $\Delta_n$ with respect to facets $\Gamma_i$ of $P_n$ containing the vertex $v_0 \in P_n$ corresponding to the monomial $a_1^2 \cdots a_{n-1}^2 \in \Delta_n$. I will give a GKZ-proof of this property and show its connection to the boundary stata in the $(n-1)$-dimensional toric Losev-Manin moduli space $\overline{L_n}$. Some variations on the above statements will be discussed in connection to the toric moduli space associated with the root system of type $B_n$ and to the mirror symmetry for $3$-dimensional cyclic quotient singularities ${\mathbb C}^3/\mu_{2n+1}$.