scholarly journals NONCOMMUTATIVE POLYNOMIAL MAPS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250076 ◽  
Author(s):  
ANDRÉ LEROY
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Ore Extension ◽  
Linear Transformations ◽  
Frobenius Automorphism ◽  
Polynomial Maps ◽  
Noncommutative Polynomial

Polynomial maps attached to polynomials of an Ore extension are naturally defined. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a finite field 𝔽q[t;θ], where θ is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring 𝔽q[x].

GAUSSIAN LAWS ON DRINFELD MODULES

2009 ◽  
Vol 05 (07) ◽  
pp. 1179-1203 ◽  
Author(s):  
WENTANG KUO ◽  
YU-RU LIU
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Valuation Ring ◽  
Finite Extension ◽  
Drinfeld Modules ◽  
Good Reduction ◽  
Prime Divisors ◽  
Frobenius Automorphism ◽  
Tate Module ◽  
Rank 2

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

K3 of truncated polynomial rings over fields of characteristic two

1987 ◽  
Vol 101 (3) ◽  
pp. 509-521 ◽  
Author(s):  
Janet Aisbett ◽  
Victor Snaith
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Witt Vectors ◽  
Characteristic Two

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).

Solvable and Nilpotent Subgroups of GL(n,qm)

1982 ◽  
Vol 34 (5) ◽  
pp. 1097-1111 ◽  
Author(s):  
Thomas R. Wolf
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Solvable Subgroup ◽  
Linear Transformations ◽  
Completely Reducible ◽  
Image Position

Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show thatwhere

scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

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