It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F(x₁,...,x_n)=x-H\left(x\right):=(x₁-H₁(x₁,...,x_n),...,x_n-H_n(x₁,...,x_n))$, where $H_j$ are homogeneous polynomials of degree 3 with real coefficients (or $H_j=0$), j = 1,...,n and H’(x) is a nilpotent matrix for each $x=(x₁,...,x_n)\in {ℝ}^n$. We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $deg{F}^{-1}\le {\left(degF\right)}^{indF-1}$, where $indF:=maxind{H}^{\text{'}}\left(x\right):x\in {ℝ}^n$. Note that the above inequality is not true when the coefficients of...

For each squarefree monomial ideal $I\subset S=k[x_1,...,x_n]$, we associate a simple finite graph $G_I$ by using the first linear syzygies of $I$. The nodes of $G_I$ are the generators of $I$, and two vertices $u_i$ and $u_j$ are adjacent if there exist variables $x,y$ such that $x{u}_i=y{u}_j$. In the cases, where $G_I$ is a cycle or a tree, we show that $I$ has a linear resolution if and only if $I$ has linear quotients and if and only if $I$ is variable-decomposable. In addition, with the same assumption on $G_I$, we characterize all squarefree monomial ideals with a...

We study $(𝒜,+,\oplus )$, the ring of arithmetical functions with unitary convolution, giving an isomorphism between $(𝒜,+,\oplus )$ and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring $(𝒜,+,·)$ of arithmetical functions with Dirichlet convolution and the power series ring $\left[[x_1,x_2,x_3,\cdots ]\right]$ on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms...

We describe the set of points over which a dominant polynomial map $f=(f_1,...,f_n):{ℂ}^n\to {ℂ}^n$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $({\prod }_{i=1}^{n}deg{f}_i-\mu \left(f\right))/\left(mi{n}_{i=1,...,n}deg{f}_i\right)$.

We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.