scholarly journals Effective results for unit equations over finitely generated integral domains

2012 ◽  
Vol 154 (2) ◽  
pp. 351-380 ◽  
Author(s):  
JAN–HENDRIK EVERTSE ◽  
KÁLMÁN GYŐRY
Keyword(s):  
Integral Domain ◽  
Linear Equations ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Polynomial Rings ◽  
Finitely Generated ◽  
Integral Domains ◽  
Finiteness Result ◽  
Unit Equations ◽  
And Function

AbstractLet A ⊃ ℤ be an integral domain which is finitely generated over ℤ and let a,b,c be non-zero elements of A. Extending earlier work of Siegel, Mahler and Parry, in 1960 Lang proved that the equation (*) aϵ +bη = c in ϵ, η ∈ A* has only finitely many solutions. Using Baker's theory of logarithmic forms, Győry proved, in 1979, that the solutions of (*) can be determined effectively if A is contained in an algebraic number field. In this paper we prove, in a quantitative form, an effective finiteness result for equations (*) over an arbitrary integral domain A of characteristic 0 which is finitely generated over ℤ. Our main tools are already existing effective finiteness results for (*) over number fields and function fields, an effective specialization argument developed by Győry in the 1980's, effective results of Hermann (1926) and Seidenberg (1974) on linear equations over polynomial rings over fields, and similar such results by Aschenbrenner, from 2004, on linear equations over polynomial rings over ℤ. We prove also an effective result for the exponential equation aγ1v1···γsvs+bγ1w1 ··· γsws=c in integers v1,…,ws, where a,b,c and γ1,…,γs are non-zero elements of A.

scholarly journals Effective results for unit points on curves over finitely generated domains

2015 ◽  
Vol 158 (2) ◽  
pp. 331-353
Author(s):  
ATTILA BÉRCZES
Keyword(s):  
Algebraic Closure ◽  
Unit Group ◽  
Number Fields ◽  
Finitely Generated ◽  
Quotient Field ◽  
Effective Version ◽  
Formula Group ◽  
Commutative Domain ◽  
Image Position ◽  
Unit Equations

AbstractLet A be a commutative domain of characteristic 0 which is finitely generated over ℤ as a ℤ-algebra. Denote by A* the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the set \begin{equation*} \mathcal{C}:=\{ (x,y)\in (A^*)^2 | F(x,y)=0 \} \end{equation*} where F(X, Y) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form XmYn - α or Xm - α Yn, with m, n ∈ ℤ⩾0, max(m, n) > 0, α ∈ K*. This result is a common generalisation of effective results of Evertse and Győry [12] on S-unit equations over finitely generated domains, of Bombieri and Gubler [5] on the equation F(x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem [20] on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.

On the Criteria of D.D. Anderson for Invertible and Flat Ideals

1986 ◽  
Vol 29 (1) ◽  
pp. 25-32 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Nonzero Ideal ◽  
Finitely Generated ◽  
Integral Domains ◽  
Maximal Ideals ◽  
Dedekind Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains ◽  
Torsion Free

AbstractLet R be an integral domain. It is proved that if a nonzero ideal I of R can be generated by n < ∞ elements, then I is invertible (i.e., flat) if and only if I(∩ Rai) = ∩ Iai for all { a1, . . ., a n﹜ ⊂ I. The article's main focus is on torsion-free R-modules E which are LCM-stable in the sense that E(Ra ∩ Rb) = Ea ∩ Eb for all a, b ∈ R. By means of linear relations, LCM-stableness is shown to be equivalent to a weak aspect of flatness. Consequently, if each finitely generated ideal of R may be 2-generated, then each LCM-stable R-module is flat. Finally, LCM-stableness of maximal ideals serves to characterize Prüfer domains, Dedekind domains, principal ideal domains, and Bézout domains amongst suitably larger classes of integral domains.

scholarly journals Elasticity in Polynomial-Type Extensions

2016 ◽  
Vol 59 (3) ◽  
pp. 581-590 ◽  
Author(s):  
Mark Batell ◽  
Jim Coykendall
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Irreducible Polynomial ◽  
Polynomial Rings ◽  
Sufficient Condition ◽  
Integral Domains ◽  
Polynomial Type ◽  
Factorial Domain ◽  
The Relationship

AbstractThe elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domain R and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every non-constant irreducible polynomial f ∈ R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.

scholarly journals Decomposition and classification of length functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1109-1129
Author(s):  
Dario Spirito
Keyword(s):  
Integral Domain ◽  
Integral Domains ◽  
Bijective Correspondence ◽  
Length Functions ◽  
Standard Representation ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

Graded integral domains in which each nonzero homogeneous t-ideal is divisorial

2019 ◽  
Vol 18 (01) ◽  
pp. 1950018 ◽  
Author(s):  
Gyu Whan Chang ◽  
Haleh Hamdi ◽  
Parviz Sahandi
Keyword(s):  
Integral Domain ◽  
Nonzero Ideal ◽  
Integral Domains ◽  
Cancellative Monoid ◽  
Integrally Closed ◽  
Graded Integral Domain

Let [Formula: see text] be a nonzero commutative cancellative monoid (written additively), [Formula: see text] be a [Formula: see text]-graded integral domain with [Formula: see text] for all [Formula: see text], and [Formula: see text]. In this paper, we study graded integral domains in which each nonzero homogeneous [Formula: see text]-ideal (respectively, homogeneous [Formula: see text]-ideal) is divisorial. Among other things, we show that if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD in which each nonzero homogeneous [Formula: see text]-ideal is divisorial if and only if each nonzero ideal of [Formula: see text] is divisorial, if and only if each nonzero homogeneous [Formula: see text]-ideal of [Formula: see text] is divisorial.

scholarly journals On the Graph of Divisibility of an Integral Domain

2015 ◽  
Vol 58 (3) ◽  
pp. 449-458 ◽  
Author(s):  
Jason Greene Boynton ◽  
Jim Coykendall
Keyword(s):  
Topological Space ◽  
Integral Domain ◽  
Point Of View ◽  
Current Understanding ◽  
Main Theme ◽  
Topological Graph ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Theoretic Point ◽  
Group Of Divisibility

AbstractIt is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

scholarly journals On the Hecke-Landau L-Series

1960 ◽  
Vol 16 ◽  
pp. 11-20 ◽  
Author(s):  
Tikao Tatuzawa
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Integral Domain ◽  
Algebraic Number Field

Let k be an algebraic number field of degree n = r1 + 2r2 with r1 real conjugates k(l) (1 ≦ l ≦ r1) and r2 pairs of complex conjugates k(m), k(m+r2)) (r1 + 1 ≦ m ≦ r1 + r2). Let o be the integral domain consisting of all integers in k.

scholarly journals A PRECISE RESULT ON THE ARITHMETIC OF NON-PRINCIPAL ORDERS IN ALGEBRAIC NUMBER FIELDS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250087 ◽  
Author(s):  
ANDREAS PHILIPP
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Theoretical Approach ◽  
Class Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Precise Result ◽  
Principal Order

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

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