Fibers of Polynomial Mappings Over Rn

斯佩特多项式的施普林格几何与两行施普林格纤维分量的舒伯特圈正性

The Springer Geometry of Specht Polynomials and Schubert cycle positivity for two row Springer fiber components

研究 A 型施普林格表示在完全旗簇同调中的几何实现，通过斯佩特多项式等进行一系列正展开，解决两行纤维的施普林格问题，证明相关猜想并识别施普林格基与网基，还推导了旗簇泊松退化轨迹分量的舒伯特圈展开。

具有一般C^2纤维的多项式是变量。I

Polynomials with general C^2-fibers are variables. I

设X'是一个3维光滑仿射代数簇，H_3（X'）=0，是一个UFD，其可逆函数是常数。假设Z是X的一个Zebriki开子集，它有一个到曲线U的态射p：Z -> U，使得p的所有纤维同构于C^2。我们证明了X'同构于C^3当且仅当X'-Z的不可约分支都不具有非孤立奇点。此外，如果X'是C^3，则p延伸到C^3上的多项式，该多项式在适当的坐标系中是线性的。结果，我们得到了在论文标题中表述的事实。

Venereau多项式及其相关的纤维丛

Venereau polynomials and related fiber bundles

在A4上的Venereau多项式v-n：=y+x^n（xz+y（yu+z^2）），n>= 1，所有纤维同构于仿射空间A3。此外，对于所有n>= 1，映射（v-n，x）：A4 -> A2产生A2上的平坦的仿射平面族。本文证明了在穿孔平面A2\0上，这个族是一个纤维丛。这个丛是平凡的当且仅当v-n是C[x]上环C[x][y，z，u]的变量。v1和v2是否是多项式环C[x，y，z，u]的变量是一个公开的问题。S. Venereau建立了v-n确实是C[x][y，z，u]在C[x]上的变量，其中n>= 3。本文在上述等价的基础上给出了Venereau结果的另一种证明。本文还讨论了含纤维仿射空间的平坦族的平凡性的其它等价性质，以及与Abhyankar-Sathaye嵌入问题和Dolgachev-Weisfeiler猜想的关系。

A computational approach to Milnor fiber cohomology

In this note we consider the Milnor fiber $F$ associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.

具有微分同胚纤维的多项式函数的分岔

Bifurcations of polynomial functions with diffeomorphic fibers

针对高维纤维下多项式分岔值的成因尚不明确的问题，构造了ℝ³上具连通纤维、近分岔值处纤维微分同胚的多项式浸没函数，还给出了ℝ²上的同类示例。

亏格1纤维化K3曲面的模空间与周期映射

Moduli spaces and period mappings of genus one fibered K3 surfaces

本文构建了多种亏格1纤维化K3曲面的模空间，建立其与局部对称空间的周期映射并计算轨形基本群，相关结果异于极化K3曲面模空间情形，可应用于K3曲面的光滑映射类群研究。

二元多项式的极小紧化

On the minimal compactification of a polynomial in two variables

考虑一个二元的本原多项式$f$，它被认为是从仿射平面到仿射直线的映射。我们研究了$f$的极小紧致化问题，从我们的结果中特别推导出：如果$f$的一根纤维在无穷远处只有一根纤维，那么$f$的所有纤维在无穷远处同时有奇点的奇点的分解.从这一点得到一个非常简单的证明铃木Abhyankar莫定理的嵌入的仿射直线在平面上。

几何单值、动机Milnor纤维的混合Hodge数与Newton多面体

Geometric monodromies, mixed Hodge numbers of motivic Milnor fibers and Newton polyhedra

本文引入多项式上同调群中局部与全局单值理论，结合环面几何与动机Milnor纤维，用Newton多面体描述等变混合Hodge数。

使用临界纤维的二元因式分解

Bivariate factorization using a critical fiber

我们推广了经典的提升和重组计划的合理和绝对因式分解的二元多项式的情况下，临界纤维。我们探索不同的策略重组的分析因素，根据复杂性的衍生物。我们表明，沿着一个关键的纤维工作，导致在某些情况下，一个良好的理论复杂性，由于重组的分析因子的数量较少。我们特别注意的情况下，多项式是非退化的P-进牛顿多面体。

On definable open continuous mappings

For a definable continuous mapping $f$ from a definable connected open subset $Ω$ of $\mathbb R^n$ into $\mathbb R^n,$ we show that the following statements are equivalent: (i) The mapping $f$ is open. (ii) The fibers of $f$ are finite and the Jacobian of $f$ does not change sign on the set of points at which $f$ is differentiable. (iii) The fibers of ${f}$ are finite and the set of points at which $f$ is not a local homeomorphism has dimension at most $n - 2.$ As an application, we prove that Whyburn's conjecture is true for definable mappings: A definable open continuous mapping of one close…

二维可积映射与有理椭圆曲面的分类

A classification of two dimensional integrable mappings and rational elliptic surfaces

通过研究有理椭圆曲面在纤维空间上的作用，对二维可积映射进行了分类。虽然QRT映射可以限制在每个纤维上，但存在几类交换纤维的可积映射。我们还证明了当广义的Heducen曲面成为指数为m的Heducen曲面时的一个等价条件。

Poincare polynomials of character varieties, Macdonald polynomials and affine Springer fibers

We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundles and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_λ[X;q,t]$ is i…

Classification at infinity of polynomials of degree 3 in 3 variables

We classify singularities at infinity of polynomials of degree 3 in 3 variables, $ f (x_0, x_1, x_2) = f_1 (x_0, x_1, x_2) + f_2 (x_0, x_1, x_2) + f_3 (x_0, x_1, x_2) $, $ f_i $ homogeneous polynomial of degree $ i $, $ i = 1,2,3 $. Based on this classification, we calculate the jump in the Milnor number of an isolated singularity at infinity, when we pass from the special fiber to a generic fiber. As an application of the results, we investigate the existence of global fibrations of degree 3 polynomials in $\mathbb{C}^3$ and search for information about the topology of the fibers in each equi…

二元复多项式的不规则纤维

Irregular fibers of complex polynomials in two variables

对于二元复多项式，研究了在其正则邻域中嵌入一个非正则纤维所诱导的同调态射，给出了这种态射是单射、满射的充要条件.特别地，这个态射是同构当且仅当对应的不规则值在无穷远处是正则的。我们将这些结果应用于消失和不变圈的研究。

具有2维纤维的实多项式映射在无穷远处的非典型值

Atypical values at infinity of real polynomial maps with $2$-dimensional fibers

本研究借助半径足够大的球面上梯度向量场的指标和刻画三变量实多项式函数无穷远非典型值，类比两变量相关结果，并给出正则纤维为2维曲面的实多项式映射的对应刻画。

奇异环面簇及附近圈层上的Milnor纤维

Milnor fibers over singular toric varieties and nearby cycle sheaves

我们提出了一个新的层理论方法计算的单值zeta函数的Milnor纤维化。作为应用，Kushnirenko和Varchenko等人关于$\CC^n$上的多项式的经典公式将推广到任何复曲面簇上的多项式函数。

复多项式映射纤维的拓扑

On the topology of fibers of complex polynomial maps

该研究综述复多项式映射纤维上同调的已知结果，建立分岔值处消失上同调新结果，推导贝蒂数上界，将相关定理推广至含任意奇点的多项式映射。

关于平面到平面的多项式映射

On polynomial mappings from the plane to the plane

设f是从平面到平面的一般多项式映射。有构造的二次型，其签名确定的数量的积极和消极的尖点的f。

Effective Whitney theorem for complex polynomial mappings of the plane

We describe the topology of a general polynomial mapping $f:\Bbb C^2\to\Bbb C^2.$

关于多项式映射的奇异簇

On a singular variety associated to a polynomial mapping

在论文“几何多项式映射在无穷远通过交叉同源性”的第二和第三作者相关联的一个给定的多项式映射$F : \C^2 \to \C^2$与非零雅可比各种其同源性或交叉同源性描述几何奇点在无穷远的映射。我们推广了这个结果。

Results on Milnor Fibrations for mixed polynomials with non-isolated singularities

In this article we investigate mixed polynomials and present conditions that can be applied on a specific class of polynomials in order to prove the existence of the Milnor Fibration, Milnor-Lê Fibration and the equivalence between them. We prove for this class the of functions that the Milnor-Lê fiber on a regular value is homeomorphic to the Milnor-Lê fiber on a critical value. We develop a criterion to verify the transversality property and apply it to a special case of the class of mixed polynomial.

Whitney theorem for complex polynomial mappings

We describe the topology of a general polynomial mapping $F=(f, g):X\to\Bbb C^2$, where $X$ is a complex plane or a complex sphere.

On quadratic polynomial mappings from the plane into the $n$ dimensional space

We show that, up to linear equivalence, there are only finitely many polynomial quadratic mappings $F:\mathbb{C}^2\to\mathbb{C}^n$ and $F:\mathbb{R}^2\to\mathbb{R}^n$. We list all possibilities.

On finite polynomial mappings

Let $X\subset \mathbb{C}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to \mathbb{C}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear mappings ${\mathcal L}( \mathbb{C}^n, \mathbb{C}^m)$ such that for every $L\in U$ the mapping $F+L$ is a finite mapping. Moreover, we can choose $U$ in this way, that all mappings $F+L; L\in U$ are topologically equivalent.

回火流与实多项式映射远纤维的上同调

Tempered currents and the cohomology of the remote fiber of a real polynomial map

设$p:R^n\to R$是多项式映射。考虑$R^n$上的回火电流的复$S'Ω^*(\RR^n)$与扭曲微分$d_p=d-dp$，其中$d$是通常的外微分，$dp$代表外乘$dp$。设$t\in R$和$F_t=p^{-1}(t)$。本文证明了当$p$是齐次的且$t$是任意正实数时，$F_t$的约化上同调$\tilda H^k(F_t;C)$同构于$H^{k+1}(S'Ω^*(\RR^n),d_p)$.我们猜想，这种同构适用于任何多项式$p$，$t$足够大（我们呼吁$F_t$为$t >> 0$远程纤维$p$），我们证明了这一猜想多项式满足一定的技术条件。其结果与A. Dimca和M. Saito等人给出了一个类似的（代数）方法来计算复多项式的一般纤维的约化上同调。

无穷远多项式映射的交同调几何

Geometry of polynomial mappings at infinity via intersection homology

对于给定的多项式映射$F:\C^2\to\C^2$与非零雅可比矩阵，我们相关联的一个品种，其同源性或相交的同源性描述的几何奇点在无穷远的这个地图。

真多项式映射的Thom猜想

The Thom Conjecture for proper polynomial mappings

设$f,g:X \to Y$是连续映射。我们说$f$拓扑等价于$g$，如果存在同胚$Φ: X\to X$和$Ψ: Y\to Y$使得$Ψ\circ f\circ Φ=g.$设$X,Y$是复光滑不可约仿射簇.证明了多项式映射的代数族$F: M\times X\ni (m, x)\mapsto F(m, x)=f_m(x)\in Y$只包含有限个拓扑不等价的真映射。特别是只有有限数量的拓扑不等价的正常多项式映射$f: \Bbb C^n\to\Bbb C^m$的有界（代数）度。这在真多项式映射的情况下给出了Thom猜想的一个肯定答案。

无穷远开折多项式映射

Unfolding polynomial maps at infinity

本文给出了在具有孤立奇点的纤维邻域内从$\C^n$到$\C$的多项式映射的拓扑模型。这是出于第一作者和李·鲁道夫早先描述的“联系的展开”。拓扑模型给出了一个有用的编码的局部和全球的monodromy的地图。新的成分-拓扑结构的奇点在无穷远的不规则纤维，编码的Milnor纤维在无穷远和单值映射这些Milnor纤维-是可恢复的n=2的全部从链接在无穷远的不规则纤维。因此，它们可以完全用拼接图来计算。我们通过计算Briancon多项式的全局单值性来说明我们的结果。

关于多项式映射的Newton非退化性

On Newton non-degeneracy of polynomial mappings

本文对一个实多项式映射或混合多项式映射$F$引入了一个新的无穷远非退化条件，它允许我们用某些牛顿多面体来逼近它的分支轨迹。本文给出了混合映射下雅可比问题的一个充分性结果。

Une stratification de Thom-Mather de lensemble asymptotique dune application polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$

We prove that the stratification of the asymptotic set associated to a polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ defined by "la méthode des façons" (arXiv:1407.5329) is a Thom-Mather stratification. Nous prouvons que la stratification de lensemble asymptotique dune application polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ définie par "la méthode des façons" (arXiv:1407.5329) est une stratification de Thom-Mather

Polynomial mappings into a Stiefel manifold and immersions

For a polynomial mapping from S^{n-k} to the Stiefel manifold \widetilde{V}_k(\R^{n}), where n-k is even, there is presented an effective method of expressing the corresponding element of the homotopy group π_{n-k}\widetilde{V}_k(\R^{n})\simeq\Z in terms of signatures of quadratic forms. There is also given a method of computing the intersection number for a polynomial immersion from S^m to \R^{2m}.

On the connectivity of Milnor fiber for mixed functions

In this note, we prove the connectivity of the Milnor fiber for a mixed polynomial $f(\mathbf z,\bar{\mathbf z})$, assuming the existence of a sequence of smooth points of $f^{-1}(0)$ converging to the origin. This result gives also a another proof for the connectivity of the Milnor fiber of a non-reduced complex analytic function which is proved by A. Dimca

可逆多项式映射的Newton非退化性

Invertible polynomial mappings via Newton non-degeneracy

本文证明了在实、复和混合多项式映射下雅可比问题的一个充分条件。这是从一个新的牛顿非退化条件下的映射的分支轨迹的研究。

复分析与几何中解析映射的刚性

On the Rigidity of Analytic Mappings in Complex Analysis and Geometry

本文研究复几何中解析映射和亚调和函数的刚性性质，通过不同条件证明了映射的拓扑度、纤维结构和极小值的存在性，揭示了李群作用的微分拓扑障碍。

Vanishing cycles, plane curve singularities, and framed mapping class groups

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_n$ and $D_n$, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of …

一个用于分类与多项式映射相关的渐近集的算法

An algorithm to classify the asymptotic set associated to a polynomial mapping

本文提出一种算法，基于“方式”定义，对度为2的三维复空间支配多项式映射的渐近集进行分类，并可推广至任意维数和次数。

平面曲线的映射类群与Meyer函数

The mapping class group and the Meyer function for plane curves

对于d ≥ 2，定义了d次平面曲线的映射类群，并证明了在这个类群上存在唯一的Meyer函数.在d=4的情况下，使用我们的Meyer函数，我们可以定义4维纤维空间的局部签名，其一般纤维是亏格3的非超椭圆紧黎曼曲面。我们的局部签名的一些计算将被给出。

具有可对角化纤维的多项式自同构的子群

Subgroups of polynomial automorphisms with diagonalizable fibers

设$R$是域$k$上的整环，$G$是$R$上多项式环$R[x_1,..., x_n]$的自同构群的子群.本文在假定$G$在$R$的分式域上可对角化的条件下，讨论了$G$何时可对角化.我们特别感兴趣的情况下，$G$是一个有限的阿贝尔群。Kraft-Russell（2014）暗示，如果$R$是$k={\bf C}$上的仿射PID，则${\rm Aut}_R(R[x_1,x_2])$的每个有限阿贝尔子群都是可对角化的。本文的主要结果之一说，这同样适用于PID $R$在任何领域$k$包含足够的单位根。

无平方因子三次多项式矩阵平方根纤维中强制交换性的多项式恢复准则

A Polynomial Recovery Criterion for Forced Commutativity in the Matrix Square-Root Fiber of a Square-Free Cubic Polynomial

研究在特征不为2和3的代数闭域\(k\)上，\(f(t)=t^3 + at + b\in k[t]\)无平方因子，\(A = f(X)\)时矩阵方程\(Y^2 = A\)，给出纤维交换性分类，即纤维非空时\(Y\)与\(X\)交换当且仅当\(X\in k[A]\)。

Lipschitz trivial values of polynomial mappings

Real or complex polynomial mappings between affines spaces admitting a Lipschitz-trivial value are completely characterized.

关于$C^2$到$C^2$的多项式映射的次数

On degree of polynomial mappings of $C^2$ to $C^2$

证明了$C^2$到$C^2$的多项式映射次数的两个结果。

一般纤维上的积分点

Integral points on generic fibers

设P（x，y）是有理多项式，Q中的k是一般值.如果曲线（P（x，y）=k）是不可约的，并且有无穷多个坐标为整数的点，则存在代数自同构，使P（x，y）成为多项式x或x^2-dy^2。此外，对于这样的曲线（和其他），我们给出了一个尖锐的界限的整数点（x，y）的数量与x和y有界。

关于多项式映射的一个不等式

An inequality for polynomial mappings

本文给出了$C^n$的多项式映射的增长性的一个估计。

Milnor纤维同调中的挠率

Torsion in Milnor fiber homology

在最近的一篇论文中，Dimca和Nemethi提出了一个问题，找到一个齐次多项式f，使得由f定义的超曲面的补的同调是无挠的，但f的Milnor纤维的同调有挠。我们证明，这确实是可能的，并通过建设表明，对于每个素数p，有一个多项式与p-扭转的同源性的米尔诺纤维。这些技术利用了超平面排列的特性。

Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves

We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.

多项式映射的另一种方法

An Alternative Approach to Polynomial Mappings

本文给出了计算代数几何中与多项式映射有关的一些重要结果，如多项式隐式化、象闭包以及多项式映射理想关系的一些性质的证明的一种简化方法。

Multiplicity of the saturated special fiber ring of height two perfect ideals

Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.

A note on some fiber-integrals

We remark that the study of a fiber-integral of the type F (s) := f =s ($ω$/df) $\land$ ($ω$/df) either in the local case where $ρ$ $\not\equiv$ 1 around 0 is C $\infty$ and compactly supported near the origin which is a singular point of {f = 0} in C n+1 , or in a global setting where f : X $\rightarrow$ D is a proper holomorphic function on a complex manifold X, smooth outside {f = 0} with $ρ$ $\not\equiv$ 1 near {f = 0}, for given holomorphic (n+1)--forms $ω$ and $ω$' , that a better control on the asymptotic expansion of F when s $\rightarrow$ 0, is obtained by using the Bernstein polynomi…

On quadratic polynomial mappings $f: \Bbb C^2 \to \Bbb C^2$

We show that up to linear equivalence, there is only finitely many polynomial quadratic mappings $f:\Bbb C^2\to\Bbb C^2$ and $f:\Bbb R^2\to \Bbb R^2.$ We list all possibilities.

热带隐式化与混合纤维多面体

Tropical Implicitization and Mixed Fiber Polytopes

软件TrIm提供了Tevelev和作者开发的热带隐含化和热带消除的实现。给定具有通用系数的多项式映射，TrIm计算图像的热带变化。当图像是超曲面时，输出是定义多项式的牛顿多面体。因此，TrIm可以用于计算混合纤维多面体，包括二级多面体。

超平面排列的Tate性质、多项式计数簇和单值性

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

Milnor纤维monodromy算子的中央超平面安排的顺序示出组合确定。特别地，给出了该单值算子平凡的一个充要条件。已知复超平面排列的补是上同调的，并且如果该排列在$\Q$上定义，则具有多项式计数。我们表明，这些属性持有相应的Milnor纤维，如果单值是平凡的。构造了一个定义在$\Q$上的超平面排列，其Milnor纤维具有非平凡的monodromy算子，上同调Tate，且没有多项式计数.这样的例子不存在于低维。

Multiplicity of the saturated special fiber ring of height three Gorenstein ideals

Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.

Filtrations on Springer fiber cohomology and Kostka polynomials

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classi…

光滑Mordell-Weil群与椭圆曲面的映射类群

The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

该研究受数论Néron-Lang定理启发，证明椭圆纤维化的光滑Mordell-Weil群有限生成，显式计算一类单连通复曲面的对应群，证明其纤维化结构拓扑同痕唯一，结合Donaldson结果得到高亏格曲面微分同胚的同痕性质。