We describe the polynomials P ∈ ℂ[x,y] such that $P(1/vⁿ,A₁vⁿ+A₂v^{2n}+...+A_{m-1}{v}^{n(m-1)}+v^{nm-k}w)\in ℂ[v,w]$. As applications we give new examples of bad field generators and examples of families of polynomials with smooth and irreducible fibers.

We present some facts, observations and remarks concerning the problem of finiteness of the rings of constants for derivations of polynomial rings over a commutative ring k containing the field ℚ of rational numbers.

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 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.

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra $K\langle x,y,z\rangle$ over a field $K$ of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of $K\langle x,y,z\rangle$. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a...