Real algebraic curves, the moment map and amoebas

Amoebas of half-dimensional varieties

An $n$-dimensional algebraic variety in $({\mathbb C}^\times)^{2n}$ covers its amoeba as well as its coamoeba generically finite-to-one. We provide an upper bound for the volume of these amoebas as well as for the number of points in the inverse images under the amoeba and coamoeba maps.

Puiseux Parametric Equations via the Amoeba of the Discriminant

Given an algebraic variety we get Puiseux type parametrizations on suitable Reinhardt domains. These domains are defined using the amoeba of hypersurfaces containing the discriminant locus of a finite projection of the variety.

Amoebas, Ronkin function and Monge-Ampère measures of algebraic curves with marked points

A generalization of the amoeba and the Ronkin function of a plane algebraic curve for a pair of harmonic functions on an algebraic curve with punctures is proposed. Extremal properties of $M$-curves are proved and connected with the spectral theory of difference operators with positive coefficients.

On the curvature of the real amoeba

For a real smooth algebraic curve $A \subset (\mathhbb{C}^*)^2$, the amoeba $\mathcal{A} \subset \mathbb{R}^2$ is the image of $A$ under the map Log : $(x,y) \mapsto (\log |x|, \log | y |)$. We describe an universal bound for the total curvature of the real amoeba $\mathcal{A}_{\mathbb{R} A}$ and we prove that this bound is reached if and only if the curve $A$ is a simple Harnack curve in the sense of Mikhalkin.

Amoebas of algebraic varieties

The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some traces of amoebas were appearing from time to time, even before the formal introduction, as auxiliary tools in several problems. After 1994 amoebas have been seen and studied in several areas of mathematics, from algebraic geometry and topology to complex analysis and combinatorics. In particular, amoebas provided a very powerful tool for studying topology of al…

On the Volume of Complex Amoebas

The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of $k$-dimensional algebraic variety in $(\mathbb{C}^*)^{n}$, with $n\geq 2k$, is finite.

阿米巴轮廓的奇点

Singularities of Amoeba Contours

该研究给出计算阿米巴轮廓奇点的显式实代数方程，分离对数临界轨迹退化与不同临界提升重合，针对平面曲线得到检测节点等的实用系统，还证明最大稀疏性不会最大化轮廓奇点。

Amoebas of algebraic varieties and tropical geometry

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content of Part 2. We pay special attention to tropical curves. Both parts also include relevant applications of the theories. Pa…

和谐阿莫埃博的热带收敛性

General tropical convergence of harmonic amoebas

本文通过施托克西统一化理论研究了和谐阿莫埃博到非简单热带曲线的收敛性，并扩展了兰格的结果以应用于更广泛的情况。

Intersections of Amoebas

Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of $n$ hypersurfaces in $(\mathbb{C}^*)^n$, which are canonical supersets of amoebas given by non-hypersurface varieties. Our main results are amoeba analogs of Bernstein's Theorem and Bézout's Theorem providing an upper bound for the number of connected compo…

Generalized logarithmic Gauss map and its relation to (co)amoebas

We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.

阿米巴轮廓的稀疏界

Sparse Bounds for Amoeba Contours

该研究推导代数曲线和超曲面的阿米巴轮廓实次数新上界，改进Lang--Shapiro--Shustin的普法夫方法，采用对数余法向消除等技术，为洛朗多项式类得出通用界和更精确稀疏估计，通过实例展示新界优于已知估计。

Amoeba finite basis does not exist in general

We show that the amoeba of a generic complex algebraic variety of codimension $1<r<n$ do not have a finite basis. In other words, it is not the intersection of finitely many hypersurface amoebas. Moreover we give a geometric characterization of the topological boundary of hypersurface amoebas refining an earlier result of F. Schroeter and T. de Wolff \cite{SW-13}.

Degenerations of amoebae and Berkovich spaces

We prove a continuity result for the fibers of the Berkovich analytification of a complex algebraic variety with respect to the the maximum of the Archimedean norm and the trivial norm. As a consequence, we obtain generalizations of a result of Mikhalkin and Rullgard about degenerations of amoebae onto tropical varieties.

Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits

We completely characterize sections of the cones of nonnegative polynomials, convex polynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial. Furthermore, nonnegativity of such polynomials $f$ coincides with solidness of the amoeba of $f$, i.e., the Log-absolute-value image of the algebraic variety $\mathcal{V}(f) \subset (\mathbb{C}^*)^n$ of $f$. These results generalize earlier works both in amoeba theory and real al…

关于阿米巴轮廓的解析性

On the Analyticity of Amoeba Contours

本文证明代数超曲面阿米巴轮廓的指数像是半代数的，并给出轮廓为实解析超曲面的判据，进而分析其内在限制，回答了关于解析性充分条件的问题。

A Nullstellensatz for amoebas

The amoeba of an affine algebraic variety V in (C^*)^r is the image of V under the map (z_1, ..., z_r) -> (log|z_1|, ..., log|z_r|). We give a characterisation of the amoeba based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of V, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension, as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hype…

Amoebas of curves and the Lyashko-Looijenga map

For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical locus $S(\mathcal{V})$ of the moment map $μ$ from $\mathcal{V}$ to its compactified amoeba $μ(\mathcal{V})$. We show that for curves $\mathcal{V}$ in a fixed complete linear system, the critical locus $S(\mathcal{V})$ is smooth apart from some real codimension $1$ walls. We then investigate the topological classification of pairs $(\mathcal{V},S(\mathcal{V}))$ when $\mathcal{V}$ and $S(\mathcal{V})$ are smooth. As a main tool, we use the Lyashko-Looijenga mapping ($\mathcal{LL}$) relative to the logarithmic Gauss map $γ:…

Amoebas, tropical varieties and compactification of Teichmuller spaces

In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We want to show how some properties of the boundary becomes straightforward if looked at from this point of view. For example…

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.

Amoebas of maximal area

To any algebraic curve A in a complex 2-torus $(\C^*)^2$ one may associate a closed infinite region in a real plane called the amoeba of A. The amoebas of different curves of the same degree come in different shapes and sizes. All amoebas in have finite area and, furthermore, there is an upper bound on the area in terms of the degree of the curve. The subject of this paper is the curves in a complex 2-torus whose amoebas are of the maximal area. We show that up to multiplication by a constant such curves are defined over real numbers and, furthermore, that their real loci are isotopic to so-ca…

On the total curvature of tropical hypersurfaces

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about thes…

Toric Cycles in the Complement of a Complex Curve in $(\mathbb{C}^{\times})^2$

The amoeba of a complex curve in the 2-dimensional complex torus is its image under the projection onto the real subspace in the logarithmic scale. The complement to an amoeba is a disjoint union of connected components that are open and convex. A toric cycle is a 2-cycle in the complement to a curve associated with a component of the complement to an amoeba. We prove homological independence of toric cycles in the complement to a complex algebraic curve with amoeba of maximal area.

Harmonic tropical morphisms and approximation

\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin's approximation…

Computing tropical curves via homotopy continuation

Exploiting a connection between amoebas and tropical curves, we devise a method for computing tropical curves using numerical algebraic geometry and give an implementation. As an application, we use this technique to compute Newton polygons of $A$-polynomials of knots.

$PSL_2$ 相位热带化导论

Introduction to $PSL_2$ phase tropicalization

本文在矩阵群 $PSL_2(\mathbb{C})$ 上引入相位热带化的非交换版本，将经典代数环面 $(\mathbb{C}^*)^n$ 中的相位热带簇方法推广到非交换情形。

Geometric and Combinatorial Structure of Hypersurface Coamoebas

Let $V$ be a complex algebraic hypersurface defined by a polynomial $f$ with Newton polytope $Δ$. It is well known that the spine of its amoeba has a structure of a tropical hypersurface. We prove in this paper that there exists a complex tropical hypersurface $V_{\infty, f}$ such that its coamoeba is homeomorphic to the closure in the real torus of the coamoeba of $V$. Moreover, the coamoeba of $V_{\infty, f}$ contains an arrangement of $(n-1)$-torus depending only on the geometry of $Δ$ and the coefficients of $f$. In addition, we can consider this arrangement, as a weighted codual hyperplan…

通过量子探针观察非对称系统中李-杨零点的代数簇

Observing Algebraic Variety of Lee-Yang Zeros in Asymmetrical Systems via a Quantum Probe

提出基于量子比特探针的非对称经典伊辛系统模拟方法，通过投影和迭代采样提取李-杨零点的代数簇，并在三量子比特NMR寄存器上实验验证，将纯数学概念引入量子模拟。

代数曲线相位所表示的链环

Links represented by phases of algebraic curves

本文证明任意环面链环可由复代数平面曲线参数映射临界值的一维连通分量并集实现，并给出牛顿多边形与分量数的显式关系，为纽结理论、热带几何和（余）阿米巴建立联系。

双周期阿兹特克二聚体模型的几何性质

Geometry of the doubly periodic Aztec dimer model

本研究针对任意参数的双周期阿兹特克菱形二聚体模型开展渐近分析，结合多领域理论框架与新提出的黎曼曲面最速下降分析方法，刻画了极限形状、北极曲线等几何特征并证明了局部涨落的收敛性。

霍奇闭链的谱分析：一种通过广义矩量法研究霍奇猜想的新途径

Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments

本研究提出研究霍奇猜想的新方法，通过推广Zernike矩定义霍奇闭链的特征形式与谱展开系数，经玩具模型验证后构建适用于K3曲面的框架，为霍奇猜想研究提供新方向。

Toric ideals, real toric varieties, and the algebraic moment map

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular, we explain the relation between linear precision and the algebraic moment map. This builds on the introduction to toric varieties by David Cox: What is a Toric Variety? at http://www.cs.amherst.edu/~dac/lectures/tutorial.ps

Algebraic polynomials and moments of stochastic integrals

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.

Algebraic certificates for the truncated moment problem

The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure on a given set K of a finite-dimensional Euclidean space. This problem has plenty of applications e.g. in optimization, control theory and statistics. When K is a compact semialgebraic set, the duality between the cone of moments of non-negative measures on K and the cone of non-negative polynomials on K yields an alternative: either y is a moment vector, or …

第12章：紧半代数集上的矩问题（修订版）

Chapter 12: The moment problem on compact semi-algebraic sets (revised version)

本文是著作第12章的修订版，为二次模和半环的Archimedean Positivstellensätze提出统一方法，并补充了半环Positivstellensätze与矩问题的新结果。

A connectedness property of algebraic moment maps

Let a connected reductive group G act on the smooth connected variety X. The cotangent bundle of X is a Hamiltonian G-variety. We show that its "total moment map" has connected fibers. This is an expanded version of section 6 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds.

On fibers of algebraic invariant moment maps

In this paper we study some properties of fibers of the invariant moment map for a Hamiltonian action of a reductive group on an affine symplectic varieity. We prove that all fibers have equal dimension. Further, under some additional restrictions, we show that the quotients of fibers are irreducible normal schemes. This paper is an expanded version of Section 5 and Subsection 7.2 of AG/0612559v1.

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree One

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

局部像矩映射那样重构具有指定像、且与到1维实仿射空间的典范投射复合的实代数映射

Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space

本研究提出一类局部类似矩映射、原像为单点集或球面乘积的非正余维数显式实代数映射，补充了Nash-Tognoli理论下显式实代数映射实例不足的问题，并给出其与典范投射复合的显式函数族。

非负多项式与代数曲线上的矩问题

Nonnegative Polynomials and Moment Problems on Algebraic Curves

本文研究了代数曲线上的非负多项式锥的面结构及其对偶矩锥的Carathéodory数，揭示了拓扑对四次规则复杂性的影响，并应用于截断矩问题。

类矩映射与具有指定原像的实代数函数

Moment-like maps and real algebraic functions with prescribed preimages

本文研究构造具有指定原像和单一奇异值的显式实代数函数，推广球面投影与Morse-Bott函数，并首次考虑原像拓扑，推动实代数奇点理论研究。

半代数集的横截性定理及其在二阶矩信号恢复与冷冻电镜中的应用

A transversality theorem for semi-algebraic sets with application to signal recovery from the second moment and cryo-EM

本文证明了半代数集在群表示中的横截性定理，并将其应用于从二阶矩恢复信号的问题，特别针对冷冻电镜推导出分子结构恢复的显式界，同时给出Gram矩阵分解、多参考对齐、相位恢复及置换不变分离器设计的信息论界。

Algebraic Identifiability of Gaussian Mixtures

We prove that all moment varieties of univariate Gaussian mixtures have the expected dimension. Our approach rests on intersection theory and Terracini's classification of defective surfaces. The analogous identifiability result is shown to be false for mixtures of Gaussians in dimension three and higher. Their moments up to third order define projective varieties that are defective. Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem for Veronese varieties to the Gaussian setting.

Some algebraic and arithmetic properties of Feynman diagrams

This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagram…

An Algebraic Geometry Method for Calculating DOS for 2D tight binding models

An algebraic geometry method is used to calculate the moments of the electron density of states as a function of the energy for lattices in the tight binding approximation. Interpreting the moments as the Mellin transform of the density allows writing down a formula for the density as an inverse Mellin transform. The method is illustrated by working out the density function for the two-dimensional square and honeycomb lattices.

Third-Order Moment Varieties of Linear Non-Gaussian Graphical Models

In this paper we study linear non-Gaussian graphical models from the perspective of algebraic statistics. These are acyclic causal models in which each variable is a linear combination of its direct causes and independent noise. The underlying directed causal graph can be identified uniquely via the set of second and third order moments of all random vectors that lie in the corresponding model. Our focus is on finding the algebraic relations among these moments for a given graph. We show that when the graph is a polytree these relations form a toric ideal. We construct explicit trek-matrices a…

乘积混合分布的矩簇

Moment Varieties for Mixtures of Products

本文在非参数代数统计框架下研究条件独立混合分布的矩簇，将其刻画为环面簇的割线簇，并重点分析其维数与定义多项式。

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group $G$ on an irreducible normal affine Poisson variety $X$. We study the invariant moment map $ψ_{G,X}:X\to \g$, that is, the composition of the moment map $μ_{G,X}:X\to g:=Lie(G)$ and the quotient morphism $g\to g\quo G$. We obtain some results on the dimensions of fibers of $ψ_{G,X}$ and the corresponding morphism of quotients $X\quo G\to g\quo G$. We also study the "Stein factorisation" of $ψ_{G,X}$. Namely, let $C_{G,X}$ denote the spectrum of the integral closure of $ψ_{G,X}^*(K[g]^G)$ in $K(X)^G$. We investigate …

实约化李群上的矩映射流与GIT估计

Moment map flow on real reductive Lie groups and GIT estimates

本文从函子化和代数视角研究了实约化李群作用下的矩映射、Kirwan-Ness分层及Lauret应用的性质。

K-矩问题：详尽介绍

The K-moment problem: A detailed introduction

本文详述多项式代数上K-矩问题，重点讨论紧致基本闭半代数集。通过Schmüdgen定理和Gelfand-Naimark-Segal构造，分析矩问题的几何与算子理论方法，探讨正定性与紧致性的关系。

计算矩多面体——聚焦张量、纠缠与矩阵乘法

Computing moment polytopes -- with a focus on tensors, entanglement and matrix multiplication

本文提出一种基于Franz数学描述的新算法，用于计算张量矩多面体，首次确定性地算出C^3⊗C^3⊗C^3及高概率算出C^4⊗C^4⊗C^4中的全部矩多面体，推动量子纠缠与代数复杂度研究。