arXiv subjects

Leonid Monin

Publications and source records attributed to Leonid Monin.

At least 19 recordsLinked to original sources

arXiv2026

A Riemann-Roch theorem for Frobenius quotients

We construct two families of commutative Frobenius rings: discrete and continuous Frobenius quotients $K_f$, resp. $A_g$, which are defined as the quotients of shift, resp. differential operator algebras by the annihilator of a polynomial. These constructions model the numerical $K$-rings and Chow rings of smooth complete varieties. We show that there is a naturally defined isomorphism $\mathbb{Q} K_f \cong \mathbb{Q} A_g$ playing the role of the Chern character, precisely when the polynomials satisfy a combinatorial analogue of the Hirzebruch-Riemann-Roch theorem, which states that there exists an invertible element $\mathrm{td} \in \mathbb{Q} A_g^\times$, called the Todd class, such that $f = \mathrm{td} \cdot g$. In this case we show that $K_f$ carries all the structure needed to make it a suitable model for $K$-rings of complete smooth varieties: $K_f$ is a $\lambda$-ring, has well-defined Chern classes, determinants, and satisfies a combinatorial analogue of Serre duality. We further show that the construction is functorial and obtain a combinatorial analogue of the Grothendieck-Riemann-Roch theorem. Lastly, we investigate the existence of larger families of isomorphisms between Frobenius quotients defined by the action of a power series on a generating set, such as the truncated Chern character which yields an integral isomorphism $K_f \cong A_g$. We provide numerous examples and applications: we give a new formula for computing Snapper polynomials of matroids, show that the dualizing class of $K_f$ coincides with dualizing classes of matroids and linear families of polytopes, realize $K$-rings of toric variety bundles as Frobenius quotients, and study $K$-rings of Ehrhart fans as well as exceptional isomorphisms in this setting.

math.AG↗

arXiv2026

Polyhedral models for K-theory of toric and flag varieties

In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties. In this paper we extend Pukhlikov-Khovanskii type presentation to the case of K-theory of toric and flag varieties. First, we study the Frobenius algebras obtained as quotients of the group algebra of free abelian group (possibly of infinite rank). Then we apply this construction to define a K-ring associated to a linear family of (virtual) polytopes. We study in detail two examples of such families: the family of integer (virtual) polytopes with a fixed normal fan and the family of (virtual) Gelfand-Zetlin polytopes. We show that the K-theory of toric and flag varieties can be realized as K-rings of the above families and use this to get natural set of relations in the above K-rings. Further, we describe the classes of structure sheaves of toric orbit closures and Schubert varieties in type A flag varieties. Finally, we show that our results also hold true in T-equivariant setting.

math.AG↗

arXiv2026

Vertex Posets, Monotone Path Polytopes, and Chow Polynomials

Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Bia{\l}ynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope.

math.CO↗

arXiv2025

Tropical Abel-Jacobi theory

To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by defining functorial Abel-Jacobi maps. We introduce, in particular, tropical Albanese varieties and formulate obstructions to algebraic equivalence of tropical cycles. In dimension 1, we show that this recovers the existing Abel-Jacobi theory for tropical curves. As an application, we consider the Ceresa class of a tropical curve which is defined as the image of the Ceresa cycle in an appropriate intermediate Jacobian under the Abel-Jacobi map. We give an explicit formula for this class entirely in terms of the combinatorics of the tropical curve.

math.AG↗

arXiv2025

When alcoved polytopes add

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 Minkowski sums. We nonetheless show that the Minkowski sum of a collection of alcoved polytopes is alcoved if and only if each pairwise sum is alcoved. This implies that the type fan of alcoved polytopes is determined by its two-dimensional cones. Moreover, we provide a complete characterization of when the Minkowski sum of alcoved simplices is again alcoved via a graphical criterion on pairs of ordered set partitions. Our characterization reduces to checking conditions on restricted partitions of length at most six. In particular, we show how the Minkowski sum decompositions of the two most well-known families of alcoved polytopes, the associahedron and the cyclohedron, fit in our framework. Additionally, inspired by the physical construction of one-loop scattering amplitudes, we present a new infinite family of alcoved polytopes, called $\widehat{D}_n$ polytopes. We conclude by drawing a connection to matroidal blade arrangements and the Dressian.

math.CO↗

arXiv2024

Chow theory of toric variety bundles

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 ordinary cohomology with homology-valued Minkowski weights, and identify the equivariant cohomology with cohomology-weighted piecewise polynomial functions. We describe the product structure on Minkowski weights via a fan displacement rule, and the non-equivariant limit via equivariant multiplicities. Along the way we establish relative analogues of the K\"unneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.

math.AG↗

arXiv2023

Fibered toric varieties

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 introduction to the class of fibered toric varieties. Then we use them to illustrate some known and conjectural results on topology and intersection theory of general toric variety bundles. Finally, using the language of fibered toric varieties, we compute the equivariant cohomology rings of smooth complete toric varieties.

math.AG↗

arXiv2023

The algebraic degree of sparse polynomial optimization

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 interpretation of polar degrees, a classical invariant of algebraic varieties, as well as Euclidean distance degrees. Furthermore, we prove the BKK generality of Lagrange systems in many instances.

math.AG↗

arXiv2023

A polyhedral homotopy algorithm for computing critical points of polynomial programs

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 homotopy algorithms. The correctness of our algorithm follows from intersection theoretic computations of the algebraic degree of polynomial optimization programs and relies on explicitly solving the tropicalization of a corresponding Lagrange system. We present experiments that demonstrate the superiority of our algorithm over traditional homotopy continuation algorithms.

math.OC↗

arXiv2022

Faces of Cosmological Polytopes

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 polytopes. We give a complete description of their faces, identify minimal faces that are not simplices and compute the number of faces in specific instances. In particular, we give a recursive description of the $f$-vector of cosmological polytopes of trees.

math.CO↗

arXiv2022

The Algebraic Degree of Coupled Oscillators

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 compute all solutions using numerical nonlinear algebra.

math.AG↗

arXiv2022

Generalized virtual polytopes and quasitoric manifolds

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 quasitoric manifolds.

math.AT↗

arXiv2021

Cohomology rings of quasitoric bundles

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 by the annihilator of an explicit polynomial. In this paper we generalize this construction to the case of quasitoric bundles. These are fiber bundles with generalized quasitoric manifolds as fibers. First we obtain a generalization of the BKK theorem to this case. Then we use recently obtained descriptions of the graded-commutative algebras which satisfy Poincaré duality to give a description of cohomology rings of quasitoric bundles.

math.AT↗

arXiv2021

Projective Embeddings of $\overline{M}_{0,n}$ and Parking Functions

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 $|ψ_n|:\overline{M}_{0,n}\to \mathbb{P}^{n-3}$ and the forgetful maps $π_i:\overline{M}_{0,i}\to \overline{M}_{0,i-1}$. 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 interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in $\mathbb{A}^2\times \mathbb{A}^3\cdots \times \mathbb{A}^{n-2}$) is equal to $(2(n-3)-1)!!=(2n-7)(2n-9) \cdots(5)(3)(1)$. As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.

math.AG↗

arXiv2021

Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes

Any totally positive $(k+m)\times n$ matrix induces a map $π_+$ from the positive Grassmannian ${\rm Gr}_+(k,n)$ to the Grassmannian ${\rm Gr}(k,k+m)$, whose image is the amplituhedron $\mathcal{A}_{n,k,m}$ and is endowed with a top-degree form called the canonical form ${\bfΩ}(\mathcal{A}_{n,k,m})$. This construction was introduced by Arkani-Hamed and Trnka, where they showed that ${\bfΩ}(\mathcal{A}_{n,k,4})$ encodes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. Moreover, the computation of ${\bfΩ}(\mathcal{A}_{n,k,m})$ is reduced to finding the triangulations of $\mathcal{A}_{n,k,m}$. However, while triangulations of polytopes are fully captured by their secondary polytopes, the study of triangulations of objects beyond polytopes is still underdeveloped. We initiate the geometric study of subdivisions of $\mathcal{A}_{n,k,m}$ and provide a concrete birational parametrization of fibers of $π: {\rm Gr}(k,n)\dashrightarrow {\rm Gr}(k,k+m)$. We then use this to explicitly describe a rational top-degree form $ω_{n,k,m}$ (with simple poles) on the fibers and compute ${\bfΩ}(\mathcal{A}_{n,k,m})$ as a summation of certain residues of $ω_{n,k,m}$. As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when $n-k-1=m$ (even). We show that, in this case, each fiber of $π$ is parametrized by a projective space and its volume form $ω_{n,k,m}$ has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes ${\bfΩ}(\mathcal{A}_{n,k,m})$ from $ω_{n,k,m}$. Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.

math.CO↗

arXiv2021

Gorenstein algebras and toric bundles

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 quotients of the ring of differential operators by the annihilator of an explicit polynomial (or more generally formal polynomial series). This provides a calculation of Macaulay's inverse systems for (not necessarily Artinian) algebras with Gorenstein duality. Our description generalizes previously know description of graded algebras with Gorenstein duality generated in degree 1. Our main motivation comes from the study of even degree cohomology rings. In particular, we apply our main result to compute the ring of cohomology classes of even degree of toric bundles and the ring of conditions of horospherical homogeneous spaces.

math.AC↗

arXiv2021

Cohomology rings of toric bundles and the ring of conditions

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 complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley-Reisner algebras. Sankaran and Uma generalized the "Stanley-Reisner description" to the case of toric bundles, i.e. equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the \BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion-Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety $G/B$. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties $G/B$ previously obtained by Kaveh.

math.AG↗

arXiv2021

Eigenforms of hyperelliptic curves with many automorphisms

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 hyperelliptic curves with many automorphisms. For each curve, the automorphism of maximal order acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of this action.

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