Leonid Monin
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View article: Mini-Workshop: Alcoved Polytopes in Physics and Optimization
Mini-Workshop: Alcoved Polytopes in Physics and Optimization Open
An alcoved polytope is a polytope whose facet normal are all in direction of type A roots. This fundamental class of polytopes has ample applications in for instance tropical geometry, statistics and algebra. The mini-workshop showcased va…
View article: When alcoved polytopes add
When alcoved polytopes add Open
Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots $e_i-e_j$. Unlike other prominent families of polytopes, like generalized permutahedra, alcoved polytopes are not closed under M…
View article: Chow theory of toric variety bundles
Chow theory of toric variety bundles Open
We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the ordi…
View article: Faces of cosmological polytopes
Faces of cosmological polytopes Open
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 polytop…
View article: Multigraded algebras and multigraded linear series
Multigraded algebras and multigraded linear series Open
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 f…
View article: Fibered toric varieties
Fibered toric varieties Open
A toric variety is called fibered if it can be represented as a total space of fibre bundle over toric base and with toric fiber. Fibered toric varieties form a special case of toric variety bundles. In this note we first give an introduct…
View article: The algebraic degree of sparse polynomial optimization
The algebraic degree of sparse polynomial optimization Open
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…
View article: Cohomology Rings of Toric Bundles and the Ring of Conditions
Cohomology Rings of Toric Bundles and the Ring of Conditions Open
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, 1…
View article: Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming
Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming Open
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 Sturm…
View article: A polyhedral homotopy algorithm for computing critical points of polynomial programs
A polyhedral homotopy algorithm for computing critical points of polynomial programs Open
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 e…
View article: Fibered Toric Varieties
Fibered Toric Varieties Open
A toric variety is called fibered if it can be represented as a total space of fibre bundle over toric base and with toric fiber. Fibered toric varieties form a special case of toric variety bundles. In this note we first give an introduct…
View article: Faces of Cosmological Polytopes
Faces of Cosmological Polytopes Open
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 polytop…
View article: The Algebraic Degree of Coupled Oscillators
The Algebraic Degree of Coupled Oscillators Open
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 Kho…
View article: Generalized virtual polytopes and quasitoric manifolds
Generalized virtual polytopes and quasitoric manifolds Open
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 …
View article: Cohomology rings of quasitoric bundles
Cohomology rings of quasitoric bundles Open
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 …
View article: Cohomology rings of quasitoric bundles
Cohomology rings of quasitoric bundles Open
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 c…
View article: Gorenstein algebras and toric bundles
Gorenstein algebras and toric bundles Open
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 …
View article: Eigenforms of hyperelliptic curves with many automorphisms
Eigenforms of hyperelliptic curves with many automorphisms Open
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 c…
View article: Multigraded algebras and multigraded linear series
Multigraded algebras and multigraded linear series Open
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 …
View article: Maximum Likelihood Degree, Complete Quadrics, and $\mathbb{C}^*$-Action
Maximum Likelihood Degree, Complete Quadrics, and $\mathbb{C}^*$-Action Open
We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on the variety of complete quadrics. This allows us to provide an explicit, basic, albeit of high c…
View article: Cohomology rings of toric bundles and the ring of conditions
Cohomology rings of toric bundles and the ring of conditions Open
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 ri…
View article: Maximum likelihood degree and space of orbits of a ${\mathbb C}^*$ action
Maximum likelihood degree and space of orbits of a ${\mathbb C}^*$ action Open
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 Grassmanni…
View article: Projective Embeddings of $\overline{M}_{0,n}$ and Parking Functions
Projective Embeddings of $\overline{M}_{0,n}$ and Parking Functions Open
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 $|ψ_n|:\overline{M}_{0,n}\to \mathb…
View article: Equations of $\,\overline{M}_{0,n}$
Equations of $\,\overline{M}_{0,n}$ Open
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 e…
View article: Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials
Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials Open
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 La…
View article: The Resultant of Developed Systems of Laurent Polynomials
The Resultant of Developed Systems of Laurent Polynomials Open
Let $R_Δ(f_1,\ldots,f_{n+1})$ be the {\it $Δ$-resultant} (see below) of $(n+1)$-tuple of Laurent polynomials. We provide an algorithm for computing $R_Δ$ assuming that an $n$-tuple $(f_2,\dots,f_{n+1})$ is {\it developed} (see sec.6). We p…