Related papers: Wild multidegrees of the form (d,d_2,d_3) for give…

In this note we show that the set mdeg(Aut(C^3)) mdeg(Tame(C^3)) is not empty. Moreover we show that this set has infinitely many elements. Since for the famous Nagata's example N of wild automorphism, mdeg N =(5,3,1) is an element of…

Let F=(F_1,...,F_n):C^n --> C^n be a polynomial mapping. By the multidegree of the mapping F we mean mdeg F=(deg F_1,...,deg F_n), an element of N^n. The aim of this paper is to study the following problem (especially for n=3): for which…

In this note we prove that the sequence (3,d_2,d_3), where d_3>= d_2>= 3, is the multidegee of some tame automorphism of C^3, if and only if 3|d_2 or d_3 is a linaer combination of 3 and d_2 with coefficients in N.

Let $3\leq d_1\leq d_2\leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $\frac{d_1}{\gcd(d_1,d_3)}\neq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or…

Let F=(F_1,...,F_n):C^n --> C^n be any polynomial mapping. By multidegree of F, denoted mdeg F, we call the sequence of positive integers (deg F_1,...,F_n). In this paper we addres the following problem: for which sequence (d_1,...,d_n)…

Let d_3 >= p_2 > p_1 >= 3 be integers such that p_1,p_2 are prime numbers. In this paper we show that the sequence (p_1,p_2,d_3) is the multidegree of some tame automorphisms of C^3 if and only if d_3 is in p_1*N+p_2*N, i.e. if and only if…

We prove that for every pair of positive integers a, b there exists a number c_0 such that for every c>=c_0 one can find a tame polynomial automorphism of C^3 with multidegree equal to (a,b,c).

Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $a\mid 2d$, then $(a, a+d, a+2d)\in\mdeg(\Tame(\mathbb{C}^3))$. (2) If $a\nmid 2d$, then, except for arithmetic progressions…

It is known that not each triple (d_1,d_2},d_3) of positive integers is a multidegree of a tame automorphism of C^3. In this paper we show that there is no tame automorphism of C^3 with multidegree (4,5,6). To do this we show that there is…

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of $\C^3$ is not closed, when the latter…

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \C^3\to\C^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j)\leq 2$ for $1\leq i<j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a…

Let $\MP_d$ denote the space of polynomials $f: \C \to \C$ of degree $d\geq 2$, modulo conjugation by $\Aut(\C)$. Using properties of polynomial trees (as introduced in [DM, math.DS/0608759]), we show that if $f_n$ is a divergent sequence…

We give examples of degree functions deg : R --> M, where R is a polynomial ring in 2 or 3 variables and M is either the integers or the natural numbers, whose behaviour with respect to derivations D : R --> R is pathological in the sense…

We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is…

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…

We consider graphs on monomials in $n$ variables of a fixed degree $d$ where two monomials are adjacent if and only if their least common multiple has degree $d+1$. We prove that when $n = 3$ and $d$ is divisible by $3$ as well as when…

Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T.

We prove that the closure (for the Zariski topology) of the set of polynomial automorphisms of the complex affine plane whose polydegree is (cd-1,b,a) contains all triangular automorphisms of degree cd+a, where a,b >1 and c>0 are integers…

We introduce an extension of the (tame) polynomial automorphism group over finite fields: the profinite (tame) polynomial automorphism group, which is obtained by putting a natural topology on the automorphism group. We show that most known…