# Questions tagged [smith-normal-form]

The smith-normal-form tag has no usage guidance.

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### Smith normal form and affine buildings

In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...

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### Generalized Smith Theorem for the torsion of cokernels

Let $R$ be a (commutative) domain and let $Q$ be its fraction field.
Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.
Let $I_k=(\det \...

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### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...

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### Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers?
In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...

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### Smith Normal Form of powers of a matrix

What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...

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### Smith Normal Form

Let $R=Z[x_{1},x_{2},\dots,x_{n}]$ be a multivariate polynomial ring. Is it possible to define a normal form for a general $m \times m$ matrix $M$ with entries from $R$?