On the permanent of a random symmetric matrix

Kwan, Matthew; Sauermann, Lisa

doi:10.1007/s00029-021-00730-6

Cited by 3 publications

(1 citation statement)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here we consider some questions of arithmetic statistics with matrices from M n pZ; Hq when n is fixed and H Ñ 8. We note that the dual situation, the entries of matrices are drawn from a small fixed set, for example, from t0, 1u but n Ñ 8 has recently received a lot of attention, see [3,10,15,19,20,24,25,32,36] and references therein as well as the recent survey [35]. The case we investigate here, that is, of fixed n and H Ñ 8, has also been investigated, for example, in [7-9, 16, 17] but it seems to be significantly less studied.…”

mentioning

confidence: 99%

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

Ostafe¹,

Shparlinski²

2022

Preprint

View full text Add to dashboard Cite

We consider the set M n pZ; Hq of n ˆn-matrices with integer elements of size at most H and obtain upper and lower bounds on the number of s-tuples of matrices from M n pZ; Hq, satisfying various multiplicative relations, including multiplicative dependence, commutativity and bounded generation of a subgroup of GL n pQq. These problems generalise those studied in the scalar case n " 1 by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. As a part of our method, we obtain a new upper bound on the number of matrices from M n pZ; Hq with a given characteristic polynomial f P ZrXs, which is uniform with respect to f . This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which f has to be fixed and irreducible.

show abstract

mentioning

confidence: 99%

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

Ostafe¹,

Shparlinski²

2022

Preprint

View full text Add to dashboard Cite

show abstract

Law of large numbers for permanents of random constrained matrices

Ganesan

2021

Indian J Pure Appl Math

View full text Add to dashboard Cite

Zeros of a growing number of derivatives of random polynomials with independent roots

Michelen,

2024

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Let X 1 , X 2 , … X_1,X_2,\ldots be independent and identically distributed random variables in C {\mathbb {C}} chosen from a probability measure μ \mu and define the random polynomial P n ( z ) = ( z − X 1 ) … ( z − X n ) . \begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*} We show that for any sequence k = k ( n ) k = k(n) satisfying k ≤ log ⁡ n / ( 5 log ⁡ log ⁡ n ) k \leq \log n / (5 \log \log n) , the zeros of the k k th derivative of P n P_n are asymptotically distributed according to the same measure μ \mu . This extends work of Kabluchko, which proved the k = 1 k = 1 case, as well as Byun, Lee and Reddy [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed k k case.

show abstract

On the permanent of a random symmetric matrix

Cited by 3 publications

References 33 publications

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

Law of large numbers for permanents of random constrained matrices

Zeros of a growing number of derivatives of random polynomials with independent roots

Contact Info

Product

Resources

About