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Register a new userSome observations on the properness of identity plus linear powers: part 2
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Tuyen Trung Truong
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This paper develops our previous work on properness of a class of maps related to the Jacobian conjecture. The paper has two main parts: - In part 1, we explore properties of the set of non-proper values $S_f$ (as introduced by Z. Jelonek) of these maps. In particular, using a general criterion for non-properness of these maps, we show that under a generic condition (to be precise later) $S_f$ contains $0$ if it is non-empty. This result is related to a conjecture in our previous paper. We obtain this by use of a dual set to $S_f$, particularly designed for the special class of maps. - In part 2, we use the non-properness criteria obtained in our work to construct a counter-example to the proposed proof in arXiv:2002.10249 of the Jacobian conjecture. In the conclusion, we present some comments pertaining the Jacobian conjecture and properness of polynomial maps in general.
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For $2$ vectors $x,yin mathbb{R}^m$, we use the notation $x * y =(x_1y_1,ldots ,x_my_m)$, and if $x=y$ we also use the notation $x^2=x*x$ and define by induction $x^k=x*(x^{k-1})$. We use $<,>$ for the usual inner product on $mathbb{R}^m$. For $A$ an $mtimes m$ matrix with coefficients in $mathbb{R}$, we can assign a map $F_A(x)=x+(Ax)^3:~mathbb{R}^mrightarrow mathbb{R}^m$. A matrix $A$ is Druzkowski iff $det(JF_A(x))=1$ for all $xin mathbb{R}^m$. Recently, Jiang Liu posted a preprint on arXiv asserting a proof of the Jacobian conjecture, by showing the properness of $F_A(x)$ when $A$ is Druzkowski, via some inequalities in the real numbers. In the proof, indeed Liu asserted the properness of $F_A(x)$ under more general conditions on $A$, see the main body of this paper for more detail. Inspired by this preprint, we research in this paper on the question of to what extend the above maps $F_A(x)$ (even for matrices $A$ which are not Druzkowski) can be proper. We obtain various necessary conditions and sufficient conditions for both properness and non-properness properties. A complete characterisation of the properness, in terms of the existence of non-zero solutions to a system of polynomial equations of degree at most $3$, in the case where $A$ has corank $1$, is obtained. Extending this, we propose a new conjecture, and discuss some applications to the (real) Jacobian conjecture. We also consider the properness of more general maps $xpm (Ax)^k$ or $xpm A(x^k)$. By a result of Druzkowski, our results can be applied to all polynomial self-mappings of $mathbb{C}^m$ or $mathbb{R}^m$.
In [9], Migliore, Miro-Roig and Nagel, proved that if $R = mathbb{K}[x,y,z]$, where $mathbb{K}$ is a field of characteristic zero, and $I=(L_1^{a_1},dots,L_r^{a_4})$ is an ideal generated by powers of 4 general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$. More recently, Migliore and Miro-Roig proved in [8] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjecture that the same holds for arbitrary powers. In this paper we will solve this conjecture and we will prove that if $I=(L_1^{a_1},dots,L_r^{a_r})$ is an ideal of $R$ generated by arbitrary powers of any set of general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$.
We generalize an example, due to Sylvester, and prove that any monomial of degree $d$ in $mathbb R[x_0, x_1]$, which is not a power of a variable, cannot be written as a linear combination of fewer than $d$ powers of linear forms.
By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.
We give a new method to construct linear spaces of matrices of constant rank, based on truncated graded cohomology modules of certain vector bundles as well as on the existence of graded Artinian modules with pure resolutions. Our method allows one to produce several new examples, and provides an alternative point of view on the existing ones.
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