Leonid Monin

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an ‐graded algebra , we define and study its volume function , which computes the asymptotics of the Hilbert function of . We relate the volume function to the volume of the fibers of the global Newton–Okounkov body of . Unlike the classical case of standa...

In this paper we study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one as an intersection product on a toric variety. As a corollary, we obtain a convex geometric inte...

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. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties ove...

In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that explicitly constructs an optimal start system, circumventing the typical bottleneck associated with polyhedral h...

Развивается теория многочлена объема виртуального многогранника на основе топологических свойств объединений наборов аффинных подпространств в вещественных евклидовых пространствах. Эта теория далее применяется для получения топологической версии теоремы Бернштейна-Кушниренко и описаний по Стенли-Райснеру и по Пухликову-Хованскому колец когомологий...

A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope for any Feynman diagram, i.e. an undirected graph. In this paper, we initiate a combinatorial study of these...

Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic complexity of this approximation problem. Using the theory of Khovanskii bases, we show that this number is given by the volume of a certain polytope. We also show how to compu...

In this paper we develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the BKK Theorem, the Stanley-Reisner and Pukhlikov-Khovanskii type descriptions for cohomology rings of generalized...

The classical Bernstein-Kushnirenko-Khovanskii theorem (or, the BKK theorem, for short) computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. In [PK92b], it was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the cohomology ring of a toric variety as...

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 cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients...

Any totally positive \((k+m)\times n\) matrix induces a map \(\pi _+\) from the positive Grassmannian \(\mathrm{Gr}_+(k,n)\) to the Grassmannian \(\mathrm{Gr}(k,k+m)\), whose image is the amplituhedron \(\mathcal {A}_{n,k,m}\) and is endowed with a top-degree form called the canonical form \(\varvec{\Omega }(\mathcal {A}_{n,k,m})\). This constructi...

We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing exists if and only if $A$ is Gorenstein. We give a description of algebras with Gorenstein duality as a quotien...

Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperellipt...

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an $\mathbb{N}^s$-graded algebra $A$, we define and study its volume function $F_A:\mathbb{N}_+^s\to \mathbb{R}$, which computes the asymptotics of the Hilbert function of $A$. We relate the volume function $F_A$ to the volume of the fibers of the global N...

We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfel...

Any totally positive $(k+m)\times n$ matrix induces a map $\pi_+$ from the positive Grassmannian $\Gr_+(k,n)$ to $\Gr(k,k+m)$, whose image is the amplituhedron $\A_{n,k,m}$ and is endowed with a top-degree form called the canonical form ${\bf\Omega}(\A_{n,k,m})$. This construction was introduced by Arkani-Hamed and Trnka, where they showed that ${\...

Let E1,…,Ek be a collection of linear series on an irreducible algebraic variety X over C which is not assumed to be complete or affine. That is, Ei⊂H0(X,Li) is a finite dimensional subspace of the space of regular sections of line bundles Li. Such a collection is called overdetermined if the generic systems1=…=sk=0, with si∈Ei does not have any ro...

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 ring of smooth projective toric varieties over $\mathbb{C}$ can be computed via the BKK Theorem. This complement...

We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on a smooth compact moduli space of orbits of a ${\mathbb C}^*$ action on the Lagrangian Grassmannian which we call Gaussian moduli. This allows us to provide an explicit, basic, albeit of high computational co...

The moduli space $\overline{M}_{0,n}$ may be embedded into the product of projective spaces $\mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^{n-3}$, using a combination of the Kapranov map $|\psi_n|:\overline{M}_{0,n}\to \mathbb{P}^{n-3}$ and the forgetful maps $\pi_i:\overline{M}_{0,i}\to \overline{M}_{0,i-1}$. We give an explicit c...

The moduli space M‾0,n may be embedded into the product of projective spaces P1×P2×⋯×Pn−3, using a combination of the Kapranov map |ψn|:M‾0,n→Pn−3 and the forgetful maps πi:M‾0,i→M‾0,i−1. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height n−3. We use this combinatorial int...

The space of measured laminations ML(Σ) associated to a topological surface Σ of genus g with n punctures is an integral piecewise linear manifold of real dimension 6g−6+2n. There is also a natural symplectic structure on ML(Σ) defined by Thurston. The integral and symplectic structures define a pair of measures on ML(Σ) which are known to be propo...

Let $E_1,\ldots,E_k$ be a collection of linear series on an algebraic variety $X$ over $\mathbb{C}$. That is, $E_i\subset H^0(X, \mathcal{L}_i)$ is a finite dimensional subspace of the space of regular sections of line bundles $ \mathcal{L}_i$. Such a collection is called overdetermined if the generic system \[ s_1 = \ldots = s_k = 0, \] with $s_i\...

The space of measured laminations $\mathcal{ML}(\Sigma)$ associated to a topological surface $\Sigma$ of genus $g$ with $n$ punctures is an integral piecewise linear manifold of real dimension $6g-6+2n$. There is also a natural symplectic structure on $\mathcal{ML}(\Sigma)$ defined by Thurston. The integral and symplectic structures define a pair o...

We study the moduli space \(\overline{M}_{0,n}\) of genus 0 curves with n marked points. Following Keel and Tevelev, we give explicit polynomials in the Cox ring of \(\mathbb{P}^{1} \times \mathbb{P}^{2} \times \cdots \times \mathbb{P}^{n-3}\) that, conjecturally, determine \(\overline{M}_{0,n}\) as a subscheme. Using Macaulay2, we prove that these...

Following work of Keel and Tevelev, we give explicit polynomials in the Cox ring of $\mathbb{P}^1\times\cdots\times\mathbb{P}^{n-3}$ that, conjecturally, determine $\overline{M}_{0,n}$ as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for $n=5, 6, 7, 8$. For $n \leq 6$ we give a cohomological proof that these polynom...

Let $ \mathcal{A}_1, \ldots, \mathcal{A}_k $ be finite sets in $ \mathbb{Z}^n $ and let $ Y \subset (\mathbb{C}^*)^n $ be an algebraic variety defined by a system of equations \[ f_1 = \ldots = f_k = 0, \] where $ f_1, \ldots, f_k $ are Laurent polynomials with supports in $\mathcal{A}_1, \ldots, \mathcal{A}_k$. Assuming that $ f_1, \ldots, f_k $ a...

Let $R_\Delta (f_1,\ldots,f_{n+1})$ be the {\it $\Delta$-resultant} (see below) of $(n+1)$-tuple of Laurent polynomials. We provide an algorithm for computing $R_\Delta$ assuming that an $n$-tuple $(f_2,\dots,f_{n+1})$ is {\it developed} (see sec.6). We provide a relation between the product of $f_1$ over roots of $f_2=\dots=f_{n+1}=0$ in $(\mathbb...