85,057 research outputs found
Irreducibility properties of Keller maps
Jedrzejewicz showed that a polynomial map over a field of characteristic zero
is invertible, if and only if the corresponding endomorphism maps irreducible
polynomials to irreducible polynomials. Furthermore, he showed that a
polynomial map over a field of characteristic zero is a Keller map, if and only
if the corresponding endomorphism maps irreducible polynomials to square-free
polynomials. We show that the latter endomorphism maps other square-free
polynomials to square-free polynomials as well.
In connection with the above classification of invertible polynomial maps and
the Jacobian Conjecture, we study irreducible properties of several types of
Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith,
we generalize the result of Bakalarski, that the components of cubic
homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any
field of characteristic zero) are irreducible.
Furthermore, we show that the Jacobian Conjecture can even be reduced to any
of these types with the extra condition that each affinely linear combination
of the components of the polynomial map is irreducible. This is somewhat
similar to reducing the planar Jacobian Conjecture to the so-called (planar)
weak Jacobian Conjecture by Kaliman.Comment: 22 page
Oka's conjecture on irreducible plane sextics
We partially prove and partially disprove Oka's conjecture on the fundamental
group/Alexander polynomial of an irreducible plane sextic. Among other results,
we enumerate all irreducible sextics with simple singularities admitting
dihedral coverings and find examples of Alexander equivalent Zariski pairs of
irreducible sextics.Comment: Final version accepted for publicatio
Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups
The problem of computing \emph{the exponent lattice} which consists of all
the multiplicative relations between the roots of a univariate polynomial has
drawn much attention in the field of computer algebra. As is known, almost all
irreducible polynomials with integer coefficients have only trivial exponent
lattices. However, the algorithms in the literature have difficulty in proving
such triviality for a generic polynomial. In this paper, the relations between
the Galois group (respectively, \emph{the Galois-like groups}) and the
triviality of the exponent lattice of a polynomial are investigated. The
\bbbq\emph{-trivial} pairs, which are at the heart of the relations between
the Galois group and the triviality of the exponent lattice of a polynomial,
are characterized. An effective algorithm is developed to recognize these
pairs. Based on this, a new algorithm is designed to prove the triviality of
the exponent lattice of a generic irreducible polynomial, which considerably
improves a state-of-the-art algorithm of the same type when the polynomial
degree becomes larger. In addition, the concept of the Galois-like groups of a
polynomial is introduced. Some properties of the Galois-like groups are proved
and, more importantly, a sufficient and necessary condition is given for a
polynomial (which is not necessarily irreducible) to have trivial exponent
lattice.Comment: 19 pages,2 figure
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