28,712 research outputs found
Generic extensions and generic Polynomials for multiplicative groups
Let be a finite-dimensional algebra over a finite field
and let be the multiplicative group of
. In this paper, we construct explicitly a generic Galois
-extension , where is a localized polynomial ring over
, and an explicit generic polynomial for in
parameters.Comment: To appear in J. Algebr
Generic Extensions and Generic Polynomials for Linear Algebraic Groups
We show the existence of and explicitly construct generic polynomials for
various groups, over fields of positive characteristic. The methods we develop
apply to a broad class of connected linear algebraic groups defined over finite
fields satisfying certain conditions on cohomology. In particular, we use our
techniques to study constructions for unipotent groups, certain algebraic tori,
and certain split semisimple groups. An attractive consequence of our work is
the construction of generic polynomials in the optimal number of parameters for
all cyclic 2-groups over all fields of positive characteristic. This contrasts
with a theorem of Lenstra, which states no cyclic 2-group of order has
a generic polynomial over
Automorphisms for skew PBW extensions and skew quantum polynomial rings
In this work we study the automorphisms of skew extensions and skew
quantum polynomials. We use Artamonov's works as reference for getting the
principal results about automorphisms for generic skew extensions and
generic skew quantum polynomials. In general, if we have an endomorphism on a
generic skew extension and there are some such that the
endomorphism is not zero on this elements and the principal coefficients are
invertible, then endomorphism act over as for some in the
ring of coefficients. Of course, this is valid for quantum polynomial rings,
with , as such Artamonov shows in his work. We use this result for giving
some more general results for skew extensions, using filtred-graded
techniques. Finally, we use localization for characterize some class the
endomorphisms and automorphisms for skew extensions and skew quantum
polynomials over Ore domains
Generic polynomials for cyclic function field extensions over certain finite fields
In this paper, we find all the generic polynomials for geometric
-cyclic function field extensions over the finite fields
where , prime integer such that and .Comment: 16 page
On parametric and generic polynomials with one parameter
Given fields , our results concern one parameter
-parametric polynomials over , and their relation to generic polynomials.
The former are polynomials of group which parametrize
all Galois extensions of of group via specialization of in , and
the latter are those which are -parametric for every field .
We show, for example, that being -parametric with taken to be the single
field is in fact sufficient for a polynomial to be generic. As a corollary, we obtain a complete list of
one parameter generic polynomials over a given field of characteristic 0,
complementing the classical literature on the topic. Our approach also applies
to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer
conjecture, we provide one parameter families of affine curves over number
fields, all with a rational point, but with no rational generic point
Generic Gaussian ideals
The content of a polynomial is the ideal generated by its
coefficients. Our aim here is to consider a beautiful formula of
Dedekind-Mertens on the content of the product of two polynomials, to explain
some of its features from the point of view of Cohen-Macaulay algebras and to
apply it to obtain some Noether normalizations of certain toric rings.
Furthermore, the structure of the primary decomposition of generic products is
given and some extensions to joins of toric rings are considered.Comment: 13 page
Retract rationality and Noether's problem
Let K be any field and G be a finite group. We will prove that, if K is any
field, p an odd prime number, and G is a non-abelian group of exponent p with
|G|=p^3 or p^4 satisfying [K(\zeta_p):K] <= 2, then K(G) is rational over K. We
will also show that K(G) is retract rational if G belongs to a much larger
class of p-groups. In particular, generic G-polynomials of G-Galois extensions
exist for these groups
HT90 and "simplest" number fields
A standard formula (1) leads to a proof of HT90, but requires proving the
existence of such that , so that
. We instead impose the condition (M), that taking
makes . Taking , we recover Shanks's simplest cubic
fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic
quartic fields, and a certain family of totally real cyclic extensions of
\Q(\cos(\pi/4m)) all have defining polynomials whose zeroes satisfy
\eqref{e:M}. Further investigation of (M) for leads to an elementary
algebraic construction of a 2-parameter family of octic polynomials with
"generic" Galois group . Imposing an additional algebraic condition
on these octics produces a new family of cyclic quartic extensions. This family
includes the "simplest" quartic fields and Washington's cyclic quartic fields
as special cases. We obtain more detailed results on our octics when the
parameters are algebraic integers in a number field. In particular, we identify
certain sets of special units, including exceptional sequences of 3 units, and
give some of their properties.Comment: 34 pages, no figures, uses href for internal cross-references.
Accepted for publication by the Illinois Journal of Mathematic
Fixed-point-free elements of iterated monodromy groups
The iterated monodromy group of a post-critically finite complex polynomial
of degree d \geq 2 acts naturally on the complete d-ary rooted tree T of
preimages of a generic point. This group, as well as its pro-finite completion,
act on the boundary of T, which is given by extending the branches to their
"ends" at infinity. We show that for nearly all polynomials, elements that have
fixed points on the boundary are rare, in that they belong to a set of Haar
measure zero. The exceptions are those polynomials linearly conjugate to
multiples of Chebyshev polynomials and a case that remains unresolved, where
the polynomial has a non-critical fixed point with many critical pre-images.
The proof involves a study of the finite automaton giving the action of
generators of the iterated monodromy group, and an application of a martingale
convergence theorem. Our result is motivated in part by applications to
arithmetic dynamics, where iterated monodromy groups furnish the "geometric
part" of certain Galois extensions encoding information about densities of
dynamically interesting sets of prime ideals.Comment: 30 pages, 5 figure
The geometric sieve and the density of squarefree values of invariant polynomials
We develop a method for determining the density of squarefree values taken by
certain multivariate integer polynomials that are invariants for the action of
an algebraic group on a vector space. The method is shown to apply to the
discriminant polynomials of various prehomogeneous and coregular
representations where generic stabilizers are finite. This has applications to
a number of arithmetic distribution questions, e.g., to the density of small
degree number fields having squarefree discriminant, and the density of certain
unramified nonabelian extensions of quadratic fields. In separate works, the
method forms an important ingredient in establishing lower bounds on the
average orders of Selmer groups of elliptic curves.Comment: 31 page
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