Abstract:
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 ${\mathbb C}^3/\mu_{2n+1}$.
