scholarly journals Sums of multivariate polynomials in finite subgroups

2018 ◽  
Vol 14 (02) ◽  
pp. 301-311
Author(s):  
Paolo Leonetti ◽  
Andrea Marino
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Subgroup ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Group Of Units ◽  
Finite Subgroups

Let [Formula: see text] be a commutative ring, [Formula: see text] a multivariate polynomial, and [Formula: see text] a finite subgroup of the group of units of [Formula: see text] satisfying a certain constraint, which always holds if [Formula: see text] is a field. Then, we evaluate [Formula: see text], where the summation is taken over all pairwise distinct [Formula: see text]. In particular, let [Formula: see text] be a power of an odd prime, [Formula: see text] a positive integer coprime with [Formula: see text], and [Formula: see text] integers such that [Formula: see text] divides [Formula: see text] and [Formula: see text] does not divide [Formula: see text] for all non-empty proper subsets [Formula: see text]; then [Formula: see text] where the summation is taken over all pairwise distinct [Formula: see text]th residues [Formula: see text] modulo [Formula: see text] coprime with [Formula: see text].

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

Free Subgroups in the Unit Groups of Integral Group Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1342-1352 ◽  
Author(s):  
B. Hartley ◽  
P. F. Pickel
Keyword(s):  
Sufficient Conditions ◽  
Finite Subgroup ◽  
Free Subgroup ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
Necessary And Sufficient ◽  
Free Subgroups

Let G be a group, ZG the group ring of G over the ring Z of integers, and U(ZG) the group of units of ZG. One method of investigating U(ZG) is to choose some property of groups and try to determine the groups G such that U(ZG) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(ZG) to be nilpotent (see also [7]), and the same authors have investigated when U(ZG) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(ZG) contains a free subgroup of rank 2. We conjecture that if this does not happen, then every finite subgroup of G is normal, from which various other conclusions then follow (see Lemma 4).

scholarly journals Complementation of the Jacobson group in a matrix ring

2005 ◽  
Vol 72 (2) ◽  
pp. 317-324
Author(s):  
David Dolžan
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Matrix Ring ◽  
Graded Rings ◽  
Matrix Rings ◽  
Generators And Relations ◽  
Group Of Units ◽  
The Matrix ◽  
Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

scholarly journals Definite determinantal representations of multivariate polynomials

2019 ◽  
Vol 19 (07) ◽  
pp. 2050129 ◽  
Author(s):  
Papri Dey
Keyword(s):  
Semidefinite Programming ◽  
Matrix Polynomial ◽  
Heuristic Method ◽  
Linear Matrix ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Bivariate Polynomial ◽  
Determinantal Representation ◽  
Representation Problem ◽  
Feasible Sets

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

TOWARD RANDOMIZED TESTING OF q-MONOMIALS IN MULTIVARIATE POLYNOMIALS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450016
Author(s):  
SHENSHI CHEN ◽  
YAQING CHEN ◽  
QUANHAI YANG
Keyword(s):  
Time Complexity ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Fixed Integer ◽  
Polynomial Size

Given any fixed integer q ≥ 2, a q-monomial is of the format [Formula: see text] such that 1 ≤ sj ≤ q - 1, 1 ≤ j ≤ t. q-monomials are natural generalizations of multilinear monomials. Recent research on testing multilinear monomials and q-monomials for prime q in multivariate polynomials relies on the property that Zq is a field when q ≥ 2 is prime. When q > 2 is not prime, it remains open whether the problem of testing q-monomials can be solved in some compatible complexity. In this paper, we present a randomized O*(7.15k) algorithm for testing q-monomials of degree k that are found in a multivariate polynomial that is represented by a tree-like circuit with a polynomial size, thus giving a positive, affirming answer to the above question. Our algorithm works regardless of the primality of q and improves upon the time complexity of the previously known algorithm for testing q-monomials for prime q > 7.

Subsemigroups of Stone-Čech compactifications

1994 ◽  
Vol 116 (1) ◽  
pp. 99-118 ◽  
Author(s):  
Talin Budak ◽  
Nilgün Işik ◽  
John Pym
Keyword(s):  
Positive Integer ◽  
Topological Space ◽  
Algebraic Structure ◽  
Discrete Group ◽  
Continuous Homomorphism ◽  
Discrete Space ◽  
Publication Date ◽  
Finite Subgroups ◽  
Discrete Subspace

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

scholarly journals RESTRICTIONS OF ALGEBRAIC GROUP REPRESENTATIONS TO FINITE SUBGROUPS

2002 ◽  
Vol 34 (2) ◽  
pp. 165-173
Author(s):  
HARM DERKSEN
Keyword(s):  
Algebraic Group ◽  
Short Proof ◽  
Linear Algebraic Group ◽  
Finite Subgroup ◽  
Group Representations ◽  
Rational Representation ◽  
Finite Dimensional ◽  
Finite Subgroups

Suppose that H is a finite subgroup of a linear algebraic group, G. It was proved by Donkin that there exists a finite-dimensional rational representation of G whose restriction to H is free. This paper gives a short proof of this in characteristic 0. The author also studies more closely which representations of H can appear as a restriction of G.

scholarly journals On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

2019 ◽  
Vol 12 (3) ◽  
pp. 51-68
Author(s):  
Oleg Gutik ◽  
Anatolii Savchuk
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Natural Partial Order ◽  
Group Congruence ◽  
Partial Isometries ◽  
Group Of Units ◽  
Minimum Group ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Free Semilattice

In this paper we study the structure of the monoid Iℕn ∞ of  cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞  is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U)  by the symmetric group Sn.

scholarly journals On 1-absorbing primary ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050111 ◽  
Author(s):  
Ayman Badawi ◽  
Ece Yetkin Celikel
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Primary Ideals ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

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