arXiv subjects

Vladislav Pokidkin

Publications and source records attributed to Vladislav Pokidkin.

3 recordsLinked to original sources

arXiv2026

Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants

In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose discriminant is an irreducible hypersurface. This resolves a conjecture of Esterov, and, in a sequel paper, leads to a complete description of components and codimensions for discriminants of square systems of equations.

math.AG↗

arXiv2026

Components of discriminants for systems of equations and irreducibility of determinants

The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of systems of polynomials with indeterminate coefficients and with the same number of equations and unknowns (square polynomial systems). This version is more involved in the sense that the discriminant may have several components of different dimensions. In the space of square matrices, we characterize row-generated subspaces on which the determinant is an irreducible polynomial. This allows us to resolve the Esterov conjecture for square polynomial systems whose discriminant is an irreducible hypersurface. Based on this result, we enumerate all the components and determine their dimensions and degrees for each of the three conventional ways to formalize the notion of a discriminant in this setting (mixed, Cayley, and A-discriminants) in cases of square and overdetermined systems. The proof of Esterov's conjecture and descriptions of the three types of discriminants are based on the theory of polymatroids.

math.AG↗

arXiv2026

Combinatorics behind discriminants of polynomial systems

We develop certain combinatorial tools for the study of discriminants of general systems of polynomial equations. Applying these tools in a sequel paper, we completely classify components of such discriminants, generalizing the classical results of Gelfand, Kapranov, and Zelevinsky on discriminants of one general multivariate polynomial. The developed tools are targeted at vector subspace arrangements and naturally extend to their combinatorial abstraction called polymatroids, which are the subject matter of this work. We explore relations between polymatroids and their induced matroids for bases, circuits, cycles, and rank functions. We define contractions for polymatroids corresponding to the contractions of the induced matroids. With a view towards applications to discriminants, we construct a new combinatorial structure induced by polymatroids, called BK-sets.

math.CO↗