Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-ε) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p^(1-ε) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-ε) for all N in a density one subset of the integers.
Let $n$ and $m$ be positive integers such that $n<m$. In this paper we compute the density of rectangular unimodular $n$ by $m$ matrices over the ring of algebraic integers of a number field.
Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by non-principal ideals yields simple constructions of further lattices including the Leech lattice.
In this paper, we compute the natural density of the set of k x n integer matrices that can be extended to an invertible n x n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form [O Id]. Connections with Cesaro's Theorem on the density of coprime integers and Quillen-Suslin's Theorem are also presented.
We introduce a natural generalization of the golden cryptography, which uses general unimodular matrices in place of the traditional Q-matrices, and prove that it preserves the original error correction properties of the encryption. Moreover, the additional parameters involved in generating the coding matrices make this unimodular cryptography resilient to the chosen plaintext attacks that worked against the golden cryptography. Finally, we show that even the golden cryptography is generally unable to correct double errors in the same row of the ciphertext matrix, and offer an additional check…
In this note, we compute the probability that a $k\times n$ matrix can be extended to an $n\times n$ invertible matrix over $\F_q[x]$, which turns out to be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's density theorem on the co-prime integers and its various generalizations are also presented.
In this paper we find a third order unimodular matrix, none of whose entries is $1$ or $-1$, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular. Further, we find third order square integer matrices $(a_{ij})$, none of the integers $a_{ij}$ being $1$ or $-1$, such that $\det{(a_{ij})}=k$ and $\det{(a_{ij}^3)}=k^3$, where $k$ is a nonzero integer.
Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.
For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple …
We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $\mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these poly…
Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.
We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients modulo 2 represented by {0,1}. The determinant of the above matrix is then closely related to the Thue-Morse sequence counting digits modulo 2 of binary integers.
We discuss recent quantitative results in connexion with Kronecker's theorem on the density of subgroups in R^n and with Dani and Raghavan's theorem on the density of orbits in the spaces of frames. We also propose several related problems. The case of the natural linear action of the unimodular group SL_2(Z) on the real plane is investigated more closely. We then establish an intriguing link between the configuration of (discrete) orbits of primitive points and the rate of density of dense orbits.
Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation $P(X) = 0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$ if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, we give a conjectural generalization …
The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives onl…
We proved recently (see \cite{lhgarasu}) the result on the title for odd prime divisors of such an $n.$ The result implies for many $n's$, more precisely, for an infinity of $n$'s with an arbitrary fixed number of prime divisors, the inexistence of circulant Hadamard matrices, and the inexistence of Barker sequences of length $n >13$. The proof used a result of Arasu. It turns out that there is another, shorter proof, of the more general result that includes the prime $p=2.$ This new proof is based on a result of Brock (see \cite[Theorem 3.1]{brock}), and besides that, requires just the defini…
We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.
We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebrai…
