搜索结果：Jacobian

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 sophisticated (and large) part of the group of automorphisms of the Grassmann algebra $Ł_n$. It is proved that the Jacobian group $Σ$ is a rational unipotent algebraic group. A (minimal) set of generators for the algebraic group $Σ$, its dimension and coordinates are found explicitly. In particular, for $n\geq 4$, \dim (§) = (n-1)2^{n-1} -n^2+2 if $n$ is even, (n-1)2^{n-1} -n^2+1 if $n$ is odd. The same is done for the Jacobian ascents - some natural algebraic overgroups of $Σ$. It is proved that the Jacobian map $\s \mapsto \det (\frac{\der \s (x_i)}{\der x_j})$ is surjective for odd $n$, and

A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result has finite length. An explicit upper bound is found for the length of the $A_n$-module ${}^σ P_n$ in terms of the degree ${\rm deg} (σ)$ of the Jacobian map $σ$. Analogous results are given for the Conjecture of Dixmier and the Poisson Conjecture. These results show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$, the images of the Jacobian maps, endomorphisms of the Weyl algebra $A_n$ and the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness woul

Motivated by results of Mestre and Voisin, in this note we mainly consider abelian varieties isogenous to hyperelliptic Jacobians In the first part we prove that a very general hyperelliptic Jacobian of genus $g\ge 4$ is not isogenous to a non-hyperelliptic Jacobian. As a consequence we obtain that the Intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary tells that the Jacobian of a very general $d$-gonal curve of genus $g \ge 4$ is not isogenous to a different Jacobian. In the second part we consider a closed subvariety $\mathcal Y \subset \mathcal A_g$ of the moduli space of principally polarized varieties of dimension $g\ge 3$. We show that if a very general element of $\mathcal Y$ is dominated by a hyperelliptic Jacobian, then $\dim \mathcal Y\ge 2g$. In particular, if the general element in $\mathcal Y$ is simple, its Kummer variety does not contain rational curves. Finally we show that a closed subvariety $\mathcal Y\subset \mathcal M_g$ of dimension $2g-1$ such that the Jacobian of a very general element of $\mathcal Y$ is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal loc

We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve $X$, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the Néron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of $X$ admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of $X$ and, moreover, that it can be realized as a quotient of the smooth locus of a suitable fine compactified Jacobian of the total blowup of $X$. Finally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda-Seshadri's coarse compactified Jacobian.

In this paper, we study a so-called Condition C1 and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. Main results obtained: - Stating new equivalent formulations of the Jacobian conjecture. - Formulating some generalisations of the Jacobian conjecture and giving both theoretical and experimental evidences to support them. - Showing Condition C1 holds for a generic matrix of any given rank, is an invariant for a certain group action, and Condition C2 is an invariant for cubic similarity matrices. - Giving one heuristic argument for the truth of the Jacobian Conjecture. - Giving an effective (time saving) method to check whether a given Druzkowski map satisfies the Jacobian conjecture, explaining theoretically and checking on many examples including those previously considered by other authors. - Proposing approaches toward resolving the Jacobian conjecture. Showing that a generic Druzkowski map satisfies the criteria of some of these approaches (see Theorem 1.12), and hence expecting to be able to check these approaches for a given Druzkowski map very quickly. -As an application, proposing a strategy to use cubic similarity to check

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 studying the jacobian conjecture we revisit the Newton polygon.

In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor assumption on the polarization, we show that a very general polarized hyperkähler fourfold $F$ of $K3^{[2]}$-type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold is potentially of Jacobian type. We also prove the Hodge conjecture in degree 4 for a generic $F$ of $K3^{[2]}$-type.

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of D-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some D-module theoretic invariant given by the degree of the Kashiwara operator.

In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph $X$ in the group of symmetries of the Jacobian of $X$. As a consequence we show that if a $3$-edge-connected graph $X$ admits a nonabelian semiregular group of automorphims, then the Jacobian of $X$ cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of $X$ is well-understood - it is equal to the number of spanning trees of $X$ - the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.

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.

The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $α$ be the exchange involution on $K[x,y]$: $α(x)= y$ and $α(y)= x$. An $α$-endomorphism $f$ of $K[x,y]$ is an endomorphism of $K[x,y]$ that preserves the involution $α$: $f α= αf$. It was shown that if $f$ is an $α$-endomorphism of $K[x,y]$ having a non-zero scalar Jacobian, then $f$ is invertible. Based on this, we bring more results that imply that a given endomorphism $f$ having a non-zero scalar Jacobian and additional conditions involving involutions, is invertible.

This paper addresses the efficient computation of Jacobian matrices for programs composed of sequential differentiable subprograms. By representing the overall Jacobian as a chain product of the Jacobians of these subprograms, we reduce the problem to optimizing the sequence of matrix multiplications, known as the Jacobian Matrix Chain Product problem. Solutions to this problem yield "optimal bracketings", which induce a precedence-constraint scheduling problem. We investigate the inherent parallelism in the solutions and develop a new dynamic programming algorithm as a heuristic that incorporates the scheduling. To assess its performance, we benchmark it against the global optimum, which is computed via a branch-and-bound algorithm.

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.

We present several versions of the Jacobian Conjecture in positive characteristic each of which if true would imply the Jacobian conjecture in characteristic 0. We test these characteristic p versions of the conjecture against several families of Jacobian pairs in characteristic p. Based on the results we propose a characteristic p approach to solving the Jacobian Conjecture in characteristic 0.

In Dacorogna-Moser theorem on the pullback equation $\varphi^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\, abla\varphi=f$) given in the author's paper "Dacorogna-Moser theorem on the Jacobian determinant equation with control of support", Discrete Cont. Dyn. Syst. 37 (2017), 4071-4089.

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers ${\cal C}^{[\bullet]}$ and the relative Jacobian ${\cal J}$ of a family of curves ${\cal C}/S$. As one of the applications, we show that in the case of a single curve $C$ this action induces a integral form of a Lefschetz $\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring of 0-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in $CH^*({\cal C}^{[\bullet]})$ and in $CH^*({\cal J})$ and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in math.AG/0204188. We also show that the our algebras of operators preserve the subrings of tautological cycles and act on them via some explicit differential operators.

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.

Let $n\geq 2$ and $\mathbb K $ be a number field of characteristic $0$. Jacobian Conjecture asserts for a polynomial map $\mathcal P$ from $\mathbb K ^n$ to itself, if the determinant of its Jacobian matrix is a nonzero constant in $\mathbb K $ then the inverse $\mathcal P^{-1}$ exists and is also a polynomial map. This conjecture was firstly proposed by Keller in 1939 for $\mathbb K ^n=\mathbb C^2$ and put in Smale's 1998 list of Mathematical Problems for the Next Century. This study is going to present a proof for the conjecture. Our proof is based on Dru{ż}kowski Map and Hadamard's Diffeomorphism Theorem, and additionally uses some optimization idea.

The main theorem (2.2) consists in two characterizations of isomorphisms of factorial domains in terms of prime or primary rings elements, and unramified, flat or weakly injective affine schemes morphisms. In order to apply this theorem to the famous Jacobian Conjecture, we first introduce its different versions in any characteristic (3.1), and give two reformulations of some these versions in terms of domains of positive characteristic (3.8) and finite prime fields (3.9). Finally, we deduce from the main theorem an original reformulation of the any characteristic version of the Jacobian Conjecture in terms of prime or primary rings elements (3.11).

Beauville introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system. In this article, we construct a variant of Beauville's system whose general level set is isomorphic to the complement of the `intersection' of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke.