Nguyen van Chau
We present some estimations on geometry of the exceptional value sets of non-zero constant Jacobian polynomial maps of $\C^2$ and it's components.
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Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree $p$, for $p$ a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain $p$-gonal curves.
We investigate which homogeneous polynomials are determined by their Jacobian ideals, and extend and complete previous results due to J. Carlson and Ph. Griffiths, K. Ueda and M. Yoshinaga, and A. Dimca and E. Sernesi.
Let $p$ be a real polynomial in two variables. We say that a polynomial $q$ is a real Jacobian mate of $p$ if the Jacobian determinant of the mapping $(p,q):\mathbb{R}^2\to\mathbb{R}^2$ is everywhere positive. We present a class of polynomials that do not have real Jacobian mates.
An important invariant of a polynomial $f$ is its Jacobian algebra defined by its partial derivatives. Let $f$ be invariant with respect to the action of a finite group of diagonal symmetries $G$. We axiomatically define an orbifold Jacobian $\mathbb{Z}/2\mathbb{Z}$-graded algebra for the pair $(f,G)$ and show its existence and uniqueness in the case, when $f$ is an invertible polynomial. In case when $f$ defines an ADE singularity, we illustrate its geometric meaning.
In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G'$ are paired, where $F$ is a Druzkowski map and $G'$ is a cubic homogeneous polynomial which related to $F$. Finally, we will find a more exactly bound for the degree of $F^{-1}$, where $F$ is a invertible map.
In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \ge 2$, we give the inverse polynomial map under the condition that $JH^3=0$. We shall show that ${\operatorname{deg}}(F^{-1})\leq d^k$ if $k \le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\operatorname{deg}}(F^{-1})> d…
A map $f$ from the quaternion skew field $H$ to itself, can also be thought as a transformation $f:R^4 \to R^4$. In this manuscript, the Jacobian $J(f)$ of $f$ is computed, in the case where $f$ is a quaternion polynomial. As a consequence, the Cauchy-Riemman equations for $f$ are derived. It is also shown that the Jacobian determinant of $f$ is non negative over $H$. The above commensurates well with the theory of analytic functions of one complex variable.
Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible, and furthermore $F$ is tame if the dimension $n\neq 4$.
The purpose of this review paper is the collection, systematization and discussion of recent results concerning the quantization approach to the Jacobian conjecture, as well as certain related topics.
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.
In the paper, we first classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $°_zv\leq 1$. After that, we generalize the structure of $H$ to $H=\big(H_1(x_1,x_2,\ldots,x_n),\allowbreak b_3x_3+\cdots+b_nx_n+H_2^{(0)}(x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$.
In the paper, we first classify all polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(°_yu,°_yh)\leq 3$, $H(0)=0$. Then we classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,u), h(x,y))$ in the case that $JH$ is nilpotent and $(°v(x,y,0),°h)\leq 3$, $H(0)=0$. Finally, we classify polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in certain conditions.
Let $f$ be an invertible polynomial and $G$ a group of diagonal symmetries of $f$. This note shows that the orbifold Jacobian algebra $\mathrm{Jac}(f,G)$ of $(f,G)$ defined by the authors and Elisabeth Werner in arXiv:1608.08962 is isomorphic as a $\mathbb{ZZ}/2\mathbb{ZZ}$-graded algebra to the Hochschild cohomology $\mathsf{HH}^*(\mathrm{MF}_G(f))$ of the dg-category $\mathrm{MF}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product formula of $\mathsf{HH}^*(\mathrm{MF}_G(f))$ given by Shklyarov in arXiv:1708.06030. We also discuss the relation of our previous resu…
Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conj…
In this paper, we first prove that $u,v,h$ are linearly dependent over ${\bf K}$ if $JH$ is nilpotent and $H$ has the form: $H=(u(x,y,z),v(u,h),h(x,y))$ with $H(0)=0$ or $H=(u(x,y),v(u,h),h(x,y,z))$ with $H(0)=0$. Then we classify polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(°_yu,°_yh)\leq 2$.
Integral curves and Jacobian Conjecture
本文针对雅可比猜想，基于L.M. Drużkowski的结果，结合半代数几何与向量场积分曲线的相关结论，推导出平展多项式自同态即为多项式自同构。
Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-λ)\_λ$ is a rational polynomial, and if the Jacobian J(f,g) is a nonzero constant for some polynomial g in K[x,y], then K[f,g] =K[x,y].
A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.
This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over C such that JH satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d >= 3.
In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of deg…
This article has been withdrown by the author.
The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for $C^1$ maps is explored here. Some results known in the polynomial case are extended to the $C^1$ context, and some special cases are resolved.
The Groebner basis and solution set of a polynomial system related to the Jacobian conjecture
本文针对与雅可比猜想相关的多项式系统，计算其Gröbner基并完整刻画解集，为研究该猜想提供了代数计算依据。
This paper has been withdrawn by the author due to a crucial argument error at p.10.
In the paper, we first classify all polynomial maps $H$ of the following form: $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$ with $JH$ nilpotent. After that, we generalize the structure of $H$ to $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2,H_1),\ldots,H_n(x_1,x_2,H_1)\big)$.
Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104
本文构造非正常二元多项式映射集，研究Jacobian条件，获得满足二维猜想的类，并用Newton多边形技术将复猜想证明推进到次数104，超越Moh边界。
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.
This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3.
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one unsurmountable problem remained even in the case of two variables, which has been unsolved since 1939, that is the Jacobian Conjecture. These are notes for author's lectures on the geometry and topology of polynomials and the Jacobian Conjecture delivered at the l'Université de Bordeaux I and at Osaka University in 2003.
On the Density of Polynomial Mappings Satisfying the Jacobian Conjecture
本文针对雅可比猜想这一未解决的代数几何问题，证明对任意维数$n\geq1$，存在非空Zariski稠密开集，其中所有雅可比矩阵可逆的多项式映射均为自同构，推进了该猜想的相关密度研究。
We prove that every polynomial map $(f,g):\mathbb{R}^2\to\mathbb{R}^2$ with nowhere vanishing Jacobian such that $\mathrm{deg}\, f\leq 5$, $\mathrm{deg}\,g \leq 6$ is injective.
A system of polynomial equations related to the Jacobian Conjecture
本文将 Jacobian 猜想的真伪归结为某多项式方程组是否有解，并证明其解集为零维。
A non-zero constant Jacobian polynomial map $F=(P,Q):\mathbb{C}^2 \longrightarrow \mathbb{C}^2$ has a polynomial inverse if the component $P$ is a simple polynomial, i.e. if, when $P$ extended to a morphism $p:X\longrightarrow \mathbb{P}^1$ of a compactification $X$ of $\mathbb{C}^2$, the restriction of $p$ to each irreducible component $C$ of the compactification divisor $D = X-\mathbb{C}^2$ is either degree 0 or 1.
For K a field of characteristic 0 and d any integer number greater than or equal to 2, we prove the invertibility of polynomial endomorphisms of the affine space of dimension d over K of the form F=Id+H, where each coordinate of H is the cube of a linear form and the cube of the Jacobian matrix of H is equal to zero. Our proof uses the inversion algorithm for polynomial maps presented in our previous paper. Our current result leads us to formulate a conjecture relating the nilpotency degree of the Jacobian matrix of H with the number of necessary steps in the inversion algorithm.
In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral domain $R$ of positive characteristic. In particular, we give characterizations of variables and univariate polynomials by using the terms of higher derivations of Jacobian type in the polynomial ring in two variables over a field of positive characteristic.
We prove that a polynomial map is invertible if and only if some associated differential ring homomorphism is bijective. To this end, we use a theorem of Crespo and Hajto linking the invertibility of polynomial maps with Picard-Vessiot extensions of partial differential fields, the theory of strongly normal extensions as presented by Kovacic and the characterization of Picard-Vessiot extensions in terms of tensor products given by Levelt.
Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever ϕ(p)=p for a mapping ϕof K[x,y], this ϕmust be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping ϕof C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant poly…
A matrix polynomial is a polynomial in a complex variable $λ$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(λ)$ is the curve $\{ (λ, μ) \in \mathbb{C}^2 \mid \mathrm{det}(P(λ) - μ\cdot \mathrm{Id}) = 0\}$. The set of matrix polynomials with a given spectral curve $C$ is known to be closely related to the Jacobian of $C$, provided that $C$ is smooth. We extend this result to the case when $C$ is an arbitrary nodal, possibly reducible, curve. In the latter case the set of matrix polynomials with spectral curve $C$ turns out to be naturally strat…
We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill-Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of these invariants is determined by the other two.
The non-proper value set of a nonsingular polynomial map from $\C^2$ into itself, if non-empty, must be a curve with one point at infinity.
We study meromorphic jacobian pairs, i.e., pairs of polynomials in one variable, with coefficients meromorphic series in a second variable, whose jacobian relative to the two variables depends only on the second variable. We pose two meromorphic jacobian conjectures about such pairs, one of which is in terms of an invariant of the pair which we call the beta invariant. These conjectures are shown to imply the bivariate algebraic jacobian conjecture which predicts that two bivariate polynomials generate the polynomial ring if their jacobian is a nonzero constant. As another technique for studyi…
Higher Jacobian matrix of weighted homogeneous polynomials and derivation algebras
本文证明加权齐次多项式高阶 Jacobian 矩阵极大子式所生成的理想仍为加权齐次，并据此部分回应高阶 Nash 爆破局部代数上不存在负权导子的猜想。
We obtain a structure theorem for the nonproperness set $S_f$ of a nonsingular polynomial mapping $f:\mathbb{C}^n \to \mathbb{C}^n$. Jelonek's results on $S_f$ and our result show that if $f$ is a counterexample to the Jacobian conjecture, then $S_f$ is a hypersurface such that $S_f\cap Z \neq \emptyset$, for any $Z\subset \mathbb{C}^n$ biregular to $\mathbb{C}^{n-1}$ and $Z = h^{-1}(0)$ for a polynomial submersion $h: \mathbb{C}^n \to \mathbb{C}$. Also, we present topological approaches to the Jacobian conjecture in $\mathbb{C}^n$.
We make two observations regarding the invertibility of Keller maps. i.e., polynomial maps for which the determinant of their Jacobian matrix is identically equal to 1. In our first result, we show that if P is a n-dimensional Keller map, defined over any extension of Q, then P has a polynomial inverse if and only if the range of P contains the cartesian product of n universal Hilbert sets. In our second result, we show that if P is a 2-dimensional Keller map, defined over any algebraic number field, then P is invertible on a set that contains almost all rational integers of K.
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the existence of a multiple root. We concentrate on bivariate polynomials and establish an original formula that relates the mixed discriminant of two bivariate Laurent polynomials with fixed support, with the sparse resultant of these polynomials and their toric Jacobian. This allows us to obtain a new proof for the bidegree of the mixed discriminant as well as to esta…
The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal.
We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More generally, we obtain the same result for open polynomial maps of a closed algebraic subset X of complex N-space to itself that have finite coimage, and for cofinite subsets of such an X invariant under the map. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.
We give necessary and sufficient conditions, in the form of matrix identities, for a polynomial f in C[X,Y] to be a component of a polynomial automorphism of C^2 and to be a component of a Keller polynomial mapping of C^2, respectively ((f,g) is a Keller polynomial mapping iff Jac(f,g) = 1).
Explicit Families of Hyperelliptic Curves with CM Jacobians
本研究构造了有理数域上雅可比允许复乘法的超椭圆曲线显式族，利用切比雪夫多项式与伽罗瓦覆盖的相互作用，证明雅可比的单性并确定相关CM域，为带CM的阿贝尔簇提供具体例子和构造准则。
Inverse limits of automorphisms of truncated polynomials and applications related to Jacobian conjecture
本文利用特征 $p$ 多项式环与逆极限技术，证明满足 Jacobian 条件且保持极大理想的多项式环同态必为自同构，为 Jacobian 猜想提供新结论。
Some arithmetic aspects of polynomial maps
本文综述 Jacobian 猜想相关结果，探讨 Essen-Lipton 提出的算术视角，并通过若干情形揭示其与仿射概形 $\mathbb{F}_p$-点计数问题的密切联系。
A polynomial map $F=(P,Q)\in \Z [x,y]^2$ with Jacobian $JF:=P_xQ_y-P_yQ_x\equiv 1$ has a polynomial inverse of integer coefficients if the complex plane curve P=0 has infinitely many integer points.
The classification of the nilpotent Jacobians with some structure has been an object of study because of its relationship with the Jacobian Conjecture. In this paper we classify the polynomial maps in dimension $n$ of the form $H = (u(x,y), u_2(x,y,x_3), \ldots, u_{n-1}(x,y,x_n), h(x,y))$ with $JH$ nilpotent. In addition we prove that the maps $X + H$ are invertible, which shows that for this kind of maps the Jacobian Conjecture is verified.
Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $φ: K[x, y] \to K[x, y]$ such that $φ(K[x, y]) = R$. The presence of other, equivalent, definitions of retracts provides several different methods of studying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $ K[x, y]$ up to an automorphism, and give several appli…
We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.
It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y]^2$ with $P_xQ_y-P_yQ_x \equiv 1$ has an inverse map in $\mathbb{Q}[x,y]^2$ if the fiber $P=0$ contains an infinite subset of $ d^{-1}\mathbb{Z}^2$ for an integer $d$.
We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.
There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $ $ {\em Jacobian set} -- the set of all algebra endomorphisms of a polynomial algebra with the Jacobian 1 -- the Jacobian conjecture claims that the Jacobian set is a {\em group}). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a {\em group} -- the {\em Jacobian group} $Σ$ -- a …
The aim of the paper is to prove two conjectures that the (left and right) global dimension of the algebra of polynomial integro-differential operators $\mathbb{I}_n$ and the Jacobian algebra $\mathbb{A}_n$ is equal to $n$ (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras $\mathbb{I}_n$ and $\mathbb{A}_n$ are neither left nor right Noetherian. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. It is proven that the global dimension of all prime factor algebras of the algebras $\mathbb{I}…
We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree $k$ homogeneous component of its Jacobian ideal for any $d-1\leq k\leq (n+1)(d-2)$. Our results generalize the previous result on the reconstruction of a homogeneous polynomial from its Jacobian ideal.
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