Related papers: A geometric criterion on the equality between BKK …
Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of…
A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically…
Geometry of sparse systems of polynomial equations (i.e. the ones with prescribed monomials and generic coefficients) is well studied in terms of their Newton polytopes. The results of this study are colloquially known as the…
Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes.…
The Gelfond-Khovanskii residue formula computes the sum of the values of any Laurent polynomial over solutions of a system of Laurent polynomial equations whose Newton polytopes have sufficiently general relative position. We discuss two…
The classical BKK theorem computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. It was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the…
We provide an approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools rely on the theory of Khovanskii bases, combined with toric geometry, the…
The well-known Bernstein-Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently the authors have found a far-reaching generalization of this theorem to generic systems of…
We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which…
We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with…
Let $L \subset \mathbb{C}^r \otimes \mathbb{C}[x_1^\pm, \ldots, x_n^\pm]$ be a finite dimensional subspace of vector-valued Laurent polynomials invariant under the action of torus $(\mathbb{C}^*)^n$. We study subvarieties in the torus,…
In [5], Srijuntongsiri and Vavasis propose the "Kantorovich-Test Subdivision algorithm", or KTS, which is an algorithm for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the…
The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick…
Lascoux polynomials are $K$-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux ($\mathsf{RSVT}$) rule for Lascoux polynomials and reverse semistandard Young…
The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes.Pukhlikov and the second author noticed that the cohomology…
In this sequel to arxiv:arXiv:1012.0835 we develop Bezout type theorems for semidegrees (including an explicit formula for {\em iterated semidegrees}) and an inequality for subdegrees. In addition we prove (in case of surfaces) a Bernstein…
In their recent inspiring paper Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar conjecture for the Br\'ezin-Gross-Witten…
The theory of Newton--Okounkov bodies provides direct relations and points out analogies between the theory of mixed volumes of convex bodies, on the one hand, and the intersection theories of Cartier divisors and of Shokurov $b$-divisors,…
A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial…
We prove a finiteness property of the values of the skein polynomial of homogeneous knots which allows to establish large classes of such knots to have arbitrarily unsharp Bennequin inequality (for the Thurston-Bennequin invariant of any of…