Piotr Jedrzejewicz
We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an n-1-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.
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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 are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.
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。
It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.
A Jacobian criterion for nonsingularity in mixed characteristic
本研究针对混合特征情形，提出含$p$-导子行的矩阵子式的Jacobian非奇异性判别法，引入perivation概念研究相关微分模类似物，还发展正特征$Γ$-构造的混合特征版本以给出更多非奇异性准则。
Let k be a field of characteristic zero. Let phi be a k-endomorphism of the polynomial algebra k[x_1,...,x_n]. It is known that phi is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. In this paper we show that phi satisfies the jacobian condition if and only if it maps irreducible polynomials to square-free polynomials. Therefore, the Jacobian Conjecture is equivalent to the following statement: every k-endomorphism of k[x_1,...,x_n], mapping irreducible polynomials to square-free polynomials, maps irreducible polynomials to irreducible polynomials.
On mappings with Jacobian one
研究次数至多为d且雅可比行列式为1的多项式自同构集合\(A(n, d)\)，证明其为扎里斯基闭集，其不可约分支与雅可比猜想相关，分支维数至少为\(n^2 - 1\)，在特定条件下集合\(X(n,d)\)的一般元素是雅可比猜想反例。
We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of irreducible and square-free elements of the subalgebra k[f1,...,fm]. We also discuss obtained properties in a more general setting - for subrings of unique factorization domains.
A note on homogeneous rank $2$ locally nilpotent derivations on $k[X,Y,Z]$
本文证明了任意素数 $p$ 下秩为 $2$ 且次数为 $p-2$ 的不可约齐次局部幂零导子可三角化，并描述了次数为 $pq-2$ 的不可约非可三角化导子的结构，进而给出了特定秩 $2$ 导子像理想生成元的明确描述。
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.
A minimal resolution for the Jacobian ideal of a generic curve arrangement
本文利用似然对应理论，为复射影平面中节点曲线排列的雅可比理想的第一、第二合冲模构造了显式极小生成元集，并推广到超曲面排列情形。
An organizing principle in the study of the Jacobian Conjecture
本文通过分析多项式映射的不可约分支，证明了雅可比猜想等价于要么所有映射都是自同构，要么一般映射不是自同构。
The complex moment sequence m(P) is assigned to a univalent polynomial P by the Cauchy transform of the P(D), where D is the unit disk. We establish the representation of the Jacobian det dm(P) in terms of roots of the derivative P'. Combining this result with the special decomposition for the Hurwitz determinants, we prove a formula for the Jacobian which was previously conjectured by C. Ullemar. As a consequence, we show that the boundary of the class of all locally univalent polynomials in $U$ is contained in the union of three irreducible algebraic surfaces.
On image ideals of nice and quasi-nice derivations over a UFD
本文研究了UFD上nice和quasi-nice导出在多项式环中的图像理想生成问题。
Let A be a polynomial algebra with complex coefficients. Let B be a finite extension ring of A which is also a polynomial algebra. We describe the factorisation of the Jacobian J of the extension into irreducibles. We also introduce the notion of a well-ramified extension and define its discriminant polynomial D. In the particular case where A is the ring of invariants of B under the action of a group (i.e., a Galois extension), this framework corresponds to the classical invariant theory of complex reflection groups. In the more general case of a well-ramified extension, we explain how the pa…
In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and $div D=0$ is a Mathieu subspace of $k[x, y]$.
Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric algebra of all derivations of $P_n$ \cite{Dzhuma99,UU2014-1}. Using the language of $\mathscr{L}_n$, for every derivation $D\in \mathscr{L}_n$ we define the associative algebra $A_D$, the Lie algebra $L_D$, and the left-symmetric algebra $\mathscr{L}_D$ related to the study of the Jacobian Conjecture. For every derivation $D\in \mathscr{L}_n$ there is a unique $n$-tuple $F=(f_1,f_2,\ldots,f_n)$ of elements of $P_n$ suc…
One develops {\em ab initio} the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A numerical invariant of a rational map is introduced, called the Jacobian dual rank. It is proved that a rational map in this general setup is birational if and only if the Jacobian dual rank attains its maximal possible value. Even in the "classical" case where the source variety is irreducible there is some gain for this invariant over the degree of the map as it is, on one hand, intrinsically related to natural constructions in…
We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a UFD of characteristic p>0. One of these conditions involves jacobians, and the second - some properties of factors. In the case of m=n this extends the known theorem of Nousiainen, and we obtain a new formulation of the jacobian conjecture in positive characteristic.
Nilpotent Jacobian maps in dimension three and stable tameness in block extensions
本文研究特征零域上具有幂零雅可比矩阵的多项式映射，通过几何与推导论证给出三维情形的规范形及显式逆与驯良分解，并推广到块扩展情形，利用剩余域判据证明稳定驯良性。
Derivations of plane algebroid curves
本文针对非正则不可约平面代数曲线，研究其导子的判定条件，建立了诱导导子的幂级数阶数下界，并证明了其值半群的相关结果。
Let $k$ be an arbitrary field of characteristic zero, $k[x, y]$ be the polynomial ring and $D$ a $k$-derivation of the ring $k[x, y]$. Recall that a nonconstant polynomial $F\in k[x, y]$ is said to be a Darboux polynomial of the derivation $D$ if $D(F)=λF$ for some polynomial $λ\in k[x, y]$. We prove that any two linearly independent over the field $k$ commuting $k$-derivations $D_{1}$ and $D_{2}$ of the ring $k[x, y]$ either have a common Darboux polynomial, or $D_{1}=D_{u_{1}}, D_{2}=D_{u_{2}}$ are Jacobian derivations i.e., $D_{i}(f)=\det J(u_{i}, f)$ for every $f\in k[x, y], i=1, 2,$ where…
We define {\bf primitive derivations} for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the {\bf primitive filtrations} of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration has already been studied, which generalizes the Hodge filtration introduced by K. Saito.
On the Castelnuovo-Mumford regularity of curve arrangements
该研究探讨复射影平面一般曲线构形的雅可比代数及分次导子模的Castelnuovo-Mumford正则性，提出无需拟齐次性假设的加-删型结果，拓展了相关经典结论的适用范围。
Invariant differential derivations for reflection groups in positive characteristic
本文研究了正特征下反射群的不变微分导子理论，给出了特征2下的Saito判据，并获得了SL(n,q)、GL(n,q)及其中间群的显式基。
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