Аннотация:
It is useful to consider a mermorphic map $f: X\to Y$ of complex analytic sets (spaces) as an analytic subset in $X \times Y$, which is the closure $\overline{G}_f$ of the graph of $f$. A fiber $\overline{G}_f$ over an indeterminacy point is interpreted through ‘blow up’ or ‘blow down’. The most transparent scheme for these procedures is realized for mappings that are inverses of the logarithmic Gauss mappings for hypersurfaces. In the theory of hypergeometric functions these inverses are called the Horn-Kapranov parametrizations.
In the paper, on which this talk is based, we study the Horn-Kapranov parametrizations for the classical discriminant to prove factorization identities for its ‘cut-offs’. More precisely, consider a polynomial in one variable
\begin{equation*}\label{1}
f(y)=a_0+a_1y+\ldots+a_ny^n.
\end{equation*}
Its discriminant is the irreducible polynomial $\Delta_n=\Delta_n(a_0,a_1, \ldots,a_n)$ with integer coefficients that vanishes if and only if $f$ has multiple roots. In [1] it is proved that the Newton polytope $\mathcal N(\Delta_n) \subset \mathbb R^{n+1}$ of this discriminant is combinatorially equivalent to an $(n-1)$-dimensional cube. Note that another proof of this fact was given by V. Batyrev, see [2].
We are interested in ‘cut-offs’ of the discriminant $\Delta_n$ to faces of its polytope $\mathcal N(\Delta_n)$ formed by intersections of $p$ facets. Recall that a ‘cut-off’ (restriction) of a polynomial $\Delta$ to a face $h$ of its polytope $\mathcal N(\Delta)$ is the sum of monomials of $\Delta$ whose exponents belong to $h$. In the talk we present explicit factorization formulas for ‘cut-offs’ of $\Delta_n$ to faces of $\mathcal N(\Delta_n)$. The study of such formulas is motivated by investigations of the structure of the universal algebraic function, see [3], [4]. The main point of these formulas is that these ‘cut-offs’ are products of discriminants of polynomials of degrees less than $n$.
These results are obtained together with V.A. Stepanenko and A.K. Tsikh.
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Reprint of the 1994 edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008, 523 pp.
V. Batyrev, Winter lectures in Arizona, 2004 http://swc.math.arizona.edu/oldaws/04Notes.html
I. A. Antipova, E. N. Mikhalkin, A. K. Tsikh, “Rational expressions for multiple roots of algebraic equations”, Mat. Sb., 209:10 (2018), 3–30; Sb. Math., 209:10 (2018), 1419–1444
A. I. Esterov, “Galois theory for general systems of polynomial equations”, Compos. Math., 155:2 (2019), 229–245