scholarly journals On the Logarithmic Probability That a Random Integral Ideal Is A $$ \pmb{\mathcal A}$$ -Free

2018 ◽  
pp. 249-258 ◽  
Author(s):  
Christian Huck
Keyword(s):  
Integral Ideal ◽  
Random Integral

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals Surjectivity of near-square random matrices

2019 ◽  
Vol 29 (2) ◽  
pp. 267-292
Author(s):  
Hoi. H. Nguyen ◽  
Elliot Paquette
Keyword(s):  
Random Matrices ◽  
High Probability ◽  
Sparse Matrices ◽  
Integral Lattice ◽  
Integral Matrix ◽  
Random Integral ◽  
Very High ◽  
Dependent Entries ◽  
Independent And Identically Distributed

AbstractWe show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

Random integral guiding functions with application to random differential complementarity systems

2018 ◽  
Vol 38 (1-2) ◽  
pp. 113
Author(s):  
Tran Dinh Ke ◽  
Nguyen Van Loi ◽  
Valeri Obukhovskii ◽  
Mai Quoc Vu
Keyword(s):  
Guiding Functions ◽  
Random Integral

scholarly journals Integral ∨-ideals

1981 ◽  
Vol 22 (2) ◽  
pp. 167-172 ◽  
Author(s):  
David F. Anderson
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Dedekind Domain ◽  
Integral Ideal ◽  
Quotient Field ◽  
Fractional Ideal

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

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