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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.
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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 $mathbb{R}^m$.
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.
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 $R/I$.
Parking functions are a widely studied class of combinatorial objects, with connections to several branches of mathematics. On the algebraic side, parking functions can be identified with the standard monomials of $M_n$, a certain monomial ideal in the polynomial ring $S = {mathbb K}[x_1, dots, x_n]$ where a set of generators are indexed by the nonempty subsets of $[n] = {1,2,dots,n}$. Motivated by constructions from the theory of chip-firing on graphs we study generalizations of parking functions determined by $M^{(k)}_n$, a subideal of $M_n$ obtained by allowing only generators corresponding to subsets of $[n]$ of size at most $k$. For each $k$ the set of standard monomials of $M^{(k)}_n$, denoted $text{stan}_n^k$, contains the usual parking functions and has interesting combinatorial properties in its own right. For general $k$ we show that elements of $text{stan}_n^k$ can be recovered as certain vector-parking functions, which in turn leads to a formula for their count via results of Yan. The symmetric group $S_n$ naturally acts on the set $text{stan}_n^k$ and we also obtain a formula for the number of orbits under this action. For the case of $k = n-2$ we study combinatorial interpretations of $text{stan}_n^{n-2}$ and relate them to properties of uprooted trees in terms of root degree and surface
We determine the Waring ranks of all sextic binary forms using a Geometric Invariant Theory approach. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3.
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