Skip to main content
MathOverflow

Return to Answer

added 171 characters in body
Source Link
Max Alekseyev
Max Alekseyev
  • 38.4k
  • 5
  • 84
  • 167

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 3 (Linear Recurring Sequences Over Artinian and Finite Rings) in the monograph Linear recurring sequences over rings and modules but I did not check that.

ADDED. On a brief look, it seems that the following answer to (ii) is implied by Proposition 17.2: $$s_A(p^k) \leq k \cdot \mathop{\rm deg} \mathrm{minpoly}(A).$$

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 3 (Linear Recurring Sequences Over Artinian and Finite Rings) in the monograph Linear recurring sequences over rings and modules but I did not check that.

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 3 (Linear Recurring Sequences Over Artinian and Finite Rings) in the monograph Linear recurring sequences over rings and modules but I did not check that.

ADDED. On a brief look, it seems that the following answer to (ii) is implied by Proposition 17.2: $$s_A(p^k) \leq k \cdot \mathop{\rm deg} \mathrm{minpoly}(A).$$

added 68 characters in body
Source Link
Max Alekseyev
Max Alekseyev
  • 38.4k
  • 5
  • 84
  • 167

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 5 of3 (Linear Recurring Sequences Over Artinian and Finite Rings) in the papermonograph Linear recurring sequences over rings and modulesLinear recurring sequences over rings and modules but I did not check that.

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 5 of the paper Linear recurring sequences over rings and modules but I did not check that.

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 3 (Linear Recurring Sequences Over Artinian and Finite Rings) in the monograph Linear recurring sequences over rings and modules but I did not check that.

Source Link
Max Alekseyev
Max Alekseyev
  • 38.4k
  • 5
  • 84
  • 167

The powers of a matrix form a linear recurrence sequence (with characteristic polynomial equal the minimal polynomial of the matrix). Hence, the question essentially is about the periods of such sequences in rings $\mathbb Z/p^k\mathbb Z$. There are tons of literature on this topic. In particular, your questions may be answered in Chapter 5 of the paper Linear recurring sequences over rings and modules but I did not check that.