MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Hi everyone.

I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice.

In particular, suppose my lattice vectors are $a$ and $b$ and I want to know whether $c$ (a Gaussian or Eisenstein integer) is in the lattice. The problem reduces to existence of rational integer solutions to the linear diophantine equation $ax+by=c$. Is there any known conditions to determine whether the Gaussian/Eisenstein integer solutions obtained (usual Bezout's) are rational integers?

Any help will be appreciated. Thanks!

share|cite|improve this question
    
You are probably better off writing everything in terms of a basis over $\mathbb{Z}$, expressing the condition as a matrix equation and using the Smith normal form or another of the standard methods of dealing with integer matrices. – Felipe Voloch Dec 20 '11 at 10:24
    
That is precisely what I am trying to avoid, going to the matrix setting. I want to stay in the complex plane for some further generalizations. Thanks anyway! – Manuel Loquias Dec 20 '11 at 13:06

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.