1,976 research outputs found
Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three
We study z-automorphisms of the polynomial algebra K[x,y,z] and the free
associative algebra K over a field K, i.e., automorphisms which fix the
variable z. We survey some recent results on such automorphisms and on the
corresponding coordinates. For K we include also results about the
structure of the z-tame automorphisms and algorithms which recognize z-tame
automorphisms and z-tame coordinates
Tame Automorphisms Fixing a Variable of Free Associative Algebras of Rank Three
We study automorphisms of the free associative algebra K over a field
K which fix the variable z. We describe the structure of the group of z-tame
automorphisms and derive algorithms which recognize z-tame automorphisms and
z-tame coordinates
Embeddings of curves in the plane
In this paper, we contribute toward a classification of two-variable
polynomials by classifying (up to an automorphism of ) polynomials whose
Newton polygon is either a triangle or a line segment. Our classification has
several applications to the study of embeddings of algebraic curves in the
plane. In particular, we show that for any , there is an irreducible
curve with one place at infinity, which has at least inequivalent
embeddings in . Also, upon combining our method with a well-known theorem
of Zaidenberg and Lin, we show that one can decide "almost" just by inspection
whether or not a polynomial fiber is an irreducible simply connected curve.Comment: 11 page
The strong Anick conjecture is true
Recently Umirbaev has proved the long-standing Anick conjecture, that is,
there exist wild automorphisms of the free associative algebra K 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. In particular, the two nontrivial
coordinates in the Anick automorphism are both wild. We establish a similar
result for several large classes of automorphisms of K. We also find a
large new class of wild automorphisms of K which is not covered by the
results of Umirbaev. Finally, we study the lifting problem for automorphisms
and coordinates of polynomial algebras, free metabelian algebras and free
associative algebras and obtain some interesting new results.Comment: 25 pages, corrected typos and acknowledgement
Polynomial Retracts and the Jacobian Conjecture
Let be the polynomial algebra in two variables over a field of
characteristic . A subalgebra of is called a retract if there
is an idempotent homomorphism (a {\it retraction}, or {\it projection})
such that . The presence
of other, equivalent, definitions of retracts provides several different
methods of studying them, and brings together ideas from combinatorial algebra,
homological algebra, and algebraic geometry. In this paper, we characterize all
the retracts of up to an automorphism, and give several applications
of this characterization, in particular, to the well-known Jacobian conjecture.
Notably, we prove that if a polynomial mapping of has
invertible Jacobian matrix {\it and } fixes a non-constant polynomial, then
is an automorphism
Affine varieties with equivalent cylinders
A well-known cancellation problem asks when, for two algebraic varieties
, the isomorphism of the cylinders and implies the isomorphism of and .
In this paper, we address a related problem: when the equivalence (under an
automorphism of ) of two cylinders and implies the equivalence of their bases and under an
automorphism of ? We concentrate here on hypersurfaces and show that
this problem establishes a strong connection between the Cancellation
conjecture of Zariski and the Embedding conjecture of Abhyankar and Sathaye. We
settle the problem for a large class of polynomials. On the other hand, we give
examples of equivalent cylinders with inequivalent bases (those cylinders,
however, are not hypersurfaces).
Another result of interest is that, for an arbitrary field , the
equivalence of two polynomials in variables under an automorphism of
implies their equivalence under a tame automorphism
of .Comment: 12 page
Degree estimate for commutators
Let K be a free associative algebra over a field K of characteristic 0 and
let each of the noncommuting polynomials f,g generate its centralizer in K.
Assume that the leading homogeneous components of f and g are algebraically
dependent with degrees which do not divide each other. We give a counterexample
to the recent conjecture of Jie-Tai Yu that deg([f,g])=deg(fg-gf) >
min{deg(f),deg(g)}. Our example satisfies deg(g)/2 < deg([f,g]) < deg(g) <
deg(f) and deg([f,g]) can be made as close to deg(g)/2 as we want. We obtain
also a counterexample to another related conjecture of Makar-Limanov and
Jie-Tai Yu stated in terms of Malcev - Neumann formal power series. These
counterexamples are found using the description of the free algebra K
considered as a bimodule of K[u] where u is a monomial which is not a power of
another monomial and then solving the equation [u^m,s]=[u^n,r] with unknowns
r,s in K.Comment: 18 page
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