Bernd Sturmfels, Jenia Tevelev, Josephine Yu
We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.
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We characterise Newton polytopes of nondegenerate quadratic forms and Newton polyhedra of Morse singularities.
The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When the image is a hypersurface, the output is the Newton polytope of the defining polynomial. TrIm can thus be used to compute mixed fiber polytopes, including secondary polytopes.
We study elimination theory in the context of Newton polytopes and develop its convex-geometric counterpart.
In this paper we compute the Newton polytope $\mathcal M_A$ of the Morse discriminant in the space of univariate polynomials with the given support set $A.$ Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of $\mathcal M_A.$
Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain-order polytopes as Newton-Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal …
Newton polytope of good symmetric polynomials
本文提出一类一般的对称多项式，证明其牛顿多胞体具有饱和性和整数分解性质，并统一涵盖多种已有对称多项式。
Newton numbers, vanishing polytopes and algebraic degrees
本文分类了具有消失Newton数的便利Newton多面体为$B_k$-多面体，部分解决了Arnold单调性问题，并推广Newton数至$\ell$-与$e$-Newton数，后者计算代数次数并刻画对偶缺陷集。
Newton-Okounkov polytopes of type $A$ flag varieties of small ranks arising from cluster structures
本文研究旗簇 $Fl(\mathbb{C}^4)$ 由簇结构产生的 Newton-Okounkov 多面体，给出其定义不等式并分类，证明在单模变换下等价。
Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case.
We study Newton polytopes of cluster variables in type A_n cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type A cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the …
We study Newton polytopes for cluster variables in cluster algebras $\mathcal{A}(Σ)$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed $Σ$. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if $Σ$ has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are \emph{empty}; that is, when they have no…
We confirm a conjecture of Monical, Tokcan and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.
It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton--Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article i…
Tropical geometry and its applications indicate a "theory of syzygies" over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Gröbner basis that captures the image of an ideal under the Newton polytope map. The image ${\rm New}(I)$ of a graded ideal $I$ under the Newton polytope is a graded sub-semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of ${\rm N…
The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation $ν_{max}$ c…
Chevalley Polytopes and Newton-Okounkov Bodies
构造了齐性空间射影嵌入中的Chevalley多面体，证明其在minuscule情形下为Newton-Okounkov体且Plücker坐标构成Khovanskii基，并提出一般情形的猜想与分解性质。
We determine the $166\,104$ extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with $D$-equivalence classes of regular triangulations of the $3$-dilated tetrahedron. We describe how to compute these triangulations and their $D$-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer amount of triangulations effectively, as well as devising a suitably fast algorithm for computation of a $D$-equivalence class.
Newton polytopes of dual Schubert polynomials
本文完整刻画了对偶舒伯特多项式的支撑集，为Huh等人2022年证明的M-凸性提供了初等替代证明，并进一步组合刻画了其牛顿多面体的顶点。
We characterize the cluster variables of skew-symmetrizable cluster algebras of rank 3 by their Newton polytopes. The Newton polytope of the cluster variable $z$ is the convex hull of the set of all $\mathbf{p}\in\mathbb{Z}^3$ such that the Laurent monomial ${\bf x}^{\mathbf{p}}$ appears with nonzero coefficient in the Laurent expansion of $z$ in the cluster ${\bf x}$. We give an explicit construction of the Newton polytope in terms of the exchange matrix and the denominator vector of the cluster variable. Along the way, we give a new proof of the fact that denominator vectors of non-initial c…
Double Schubert polynomials do have saturated Newton polytopes
该研究受多重次数理论启发，通过引入理想标准化概念，证明了double Schubert多项式具有饱和牛顿多面体性质，解决了相关猜想，同时表明Schubert行列式理想等的多重次数多项式支撑集为离散拟阵。
Newton-Okounkov polytopes of Schubert varieties arising from cluster structures
本文基于 cluster 代数理论，利用推广扩展 g-向量的特定赋值构造 Schubert 簇的 Newton-Okounkov 体，并揭示其与 string 及 Nakashima-Zelevinsky 多面体的联系。
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that t…
Supersymmetric Schur polynomials have saturated Newton polytopes
该研究证明每个超对称舒尔多项式具有饱和牛顿多面体（SNP），通过表格理论描述支撑并编码为带完全幺模约束矩阵的多面体，结合霍夫曼-克鲁斯卡尔准则得出整性以确立SNP性质。
A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. Fo…
We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in "Lectures on Algebraic Statistics" (Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph $K_{2,4}$ by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way …
Let $\K=\C$, or $\R$, and $S_f$ be the set of points in $\K^n$ at which a polynomial map $f:\K^n\rightarrow\K^n$ is non-proper. Jelonek proved that $S_f$ is a semi-algebraic set that is ruled by polynomial curves, with $\dim S_f\leq n-1$, and provided a method to compute $S_f$ for $\K = \C$. However, such methods do not exist for $\K = \R$. In this paper, we establish a straightforward description of $S_f$ for a large family of non-proper maps $f$ using the Newton polytopes of the polynomials appearing in $f$. Thus resulting in a new method for computing $S_f$ that works for $\K=\R$, and highl…
The Fine interior of dilations of a rational polytope
本文通过纯组合方法描述有理多面体所有膨胀的Fine内部，从而计算最小非空Fine内部膨胀，并分类了最小膨胀的Fine内部仅含一个点的空Fine内部格3-多面体。
Lattice 3-polytopes of lattice width 2 and corresponding toric hypersurfaces
本文分类了至多含40个内部格点且格宽为2的格3-多胞体的二维Fine内部，从而构造出大量亏格不超过40的一般型曲面。
Newton polytopes of dual $k$-Schur polynomials
本文通过递归生成半标准$k$-表的算法，证明以$k$-有界分拆为指标的对偶$k$-Schur多项式与同指标Schur多项式支撑集一致、共享饱和牛顿多面体，进而得到三类多项式的M-凸性。
Newton polytopes of immanants of some combinatorial matrices
本文研究了Jacobi-Trudi和Giambelli矩阵immanants的牛顿多面体，通过格路方法验证了Jacobi-Trudi矩阵在特殊情况下牛顿多面体的饱和性，并证明了Giambelli矩阵immanants的牛顿多面体具有饱和性。
On Newton polytopes of Lagrangian augmentations
本笔记利用牛顿多面体研究 Legendrian 子流形的 Lagrangian 填充，证明其可区分无穷多填充，并在有限特征下计算发现存在具无穷多不可定向精确填充的 Legendrian 链环。
Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in t…
The Newton polytope of the Kronecker product
本文研究了两个 Schur 函数的 Kronecker 积，利用 Horn 不等式证明了其单项式展开具有饱和 Newton 多面体的猜想特例，并导出了 Kronecker 系数正性的必要条件。
ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras
本文将ABHY结合多面体新构造推广至任意单标Dynkin类型的广义结合多面体，并发现该构造同样可生成对应簇代数所有 $F$-多项式的牛顿多面体，同时证明了g-向量扇相关环簇的nef锥具有单纯性。
The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 D-equivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.
The graph of implicit edge dependencies for indecomposability and beyond
本文通过引入隐式边依赖图，提出了一种新的不可分解性判定准则，统一并推广了现有方法，并应用于构造新的不可分解变形置换体及反驳Smilansky猜想。
We consider polynomially and rationally parameterized curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of its implicit polygon, i.e. of the implicit equation's Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations of point sets defined by Cayley's trick. We distinguish polynomial and rational parameterizations, where the latter may have the same or different denominators; the implicit pol…
Newton-Okounkov bodies of flag varieties and combinatorial mutations
本文利用源自Fano流形镜像对称的组合突变框架，通过迭代组合突变建立了旗簇的string polytopes、Nakashima-Zelevinsky polytopes与FFLV polytopes等Newton-Okounkov体之间的联系。
Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes
本文通过牛顿多面体研究TSSS簇代数中F-多项式的性质，提出了簇变量Laurent表达式的递推构造，证明了多面体基的强正性，并构建了关键变量间的显式映射关系。
The Newton polytope and Lorentzian property of chromatic symmetric functions
本文研究Dyck路径无差别图的色对称函数，证明其允许着色权重构成置换多面体的格点并给出支配权公式，猜想并部分证明其洛伦兹性质，同时将牛顿多面体结果推广到(3+1)-自由偏序集。
We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized vo…
Schubitopes were introduced by Monical, Tokcan and Yong as a specific family of generalized permutohedra. It was proven by Fink, Mészáros and St.$\,$Dizier that Schubitopes are the Newton polytopes of the dual characters of flagged Weyl modules. Important cases of Schubitopes include the Newton polytopes of Schubert polynomials and key polynomials. In this paper, we develop a combinatorial rule to generate the vertices of Schubitopes. As an application, we show that the vertices of the Newton polytope of a key polynomial can be generated by permutations in a lower interval in the Bruhat order,…
Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove tha…
We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of the authors, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. …
The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class $NP\cap coNP$ of problems with "good characterizations". This suggests a new algebraic combinatorics viewpoint on complexity theory. This report discusses the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a …
Basecondary polytopes
本文引入basecondary polytopes类多胞体，可统一研究次级多胞体、基多胞体及牛顿多胞体等，还涵盖无法归约为传统A-判别式与次级多胞体的Lyashko–Looijenga映射判别式，为相关几何研究提供统一框架。
We use the stabilization functors to study the combinatorial aspects of the $F$-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the $F$-polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the $F$-polynomial is saturated for any rigid representation. We provide many examples and counterexamples, and pose several conjectur…
Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope
本文为代数超曲面引入 Newton-Hironaka 多面体的扰动变体，由实数参数化，在参数值合适时接近原多面体，并提供不依赖参数选择的额外信息。
Newton polytopes of fireworks Grothendieck polynomials
本文证明烟花排列的 Grothendieck 多项式支撑集达到最大，并给出其齐次化具有 M-凸性，且存在分层排列使其支撑集包含原支撑集。
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial time algorithm for decomposing polygons. For higher dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithm include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.
We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to 1. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials…
Mirković-Vilonen Polytopes from Combinatorics
本文从组合学角度研究 Mirković-Vilonen 多胞体，给出拟阵多胞体、旗拟阵多胞体、Bruhat 区间多胞体和 Schubitope 成为 MV 多胞体的分类与组合条件，并证明 Schubert 多项式与 key 多项式的 Newton 多胞体均为 MV 多胞体。
Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results: we show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of Fujita approximation theorem for …
We study a family of dissections of flow polytopes arising from the subdivision algebra. To each dissection of a flow polytope, we associate a polynomial, called the left-degree polynomial, which we show is invariant of the dissection considered (proven independently by Grinberg). We prove that left-degree polynomials encode integer points of generalized permutahedra. Using that certain left-degree polynomials are related to Grothendieck polynomials, we resolve special cases of conjectures by Monical, Tokcan, and Yong regarding the saturated Newton polytope property of Grothendieck polynomials…
The concept of sums of nonnegative circuit polynomials (SONC) was recently introduced as a new certificate of nonnegativity especially for sparse polynomials. In this paper, we explore the relationship between nonnegative polynomials and SONC polynomials. As a first result, we provide sufficient conditions for nonnegative polynomials with general Newton polytopes to be SONC polynomials, which generalizes the previous result on nonnegative polynomials with simplex Newton polytopes. Secondly, we prove that every SONC polynomial admits a SONC decomposition without cancellation. In other words, SO…
Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton's conjecture.
It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. In this paper, we continue this study by describing the c- and g-vectors, and by providing a conjectured description of the Newton polytopes of the F-polynomials. In particular, we show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F-polynomials, or of the cluster variables, respectively.
Saturation for Non-Symmetric Macdonald Polynomials
本文证明非对称Macdonald多项式支持集为M-凸集，解决了Monical等人2019年的猜想，并展示了仿射Demazure特征在GL类型下具有M-凸支持，从而回答了Besson和Hong的2022年问题。
On the discrete Heine-Shephard problem for four lattice polygons
本文研究了四个格子多边形的离散Heine-Shephard问题，揭示了格子多面体混合面积上的额外算术约束导致平方自由多项式刻画失效的现象。
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