# Tagged Questions

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### Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation. I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
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### Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows: For a ...
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### Linear system with many solutions from a finite set

Basically I am looking for a linear system with many solutions from a finite set. Choose a finite set of rationals $S$ and fix positive integer $k$. Let $A$ be a linear system with $n$ variables ...
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### A lower bound on the number of matrices whose image contains all multiples of $p^e$

Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that ...
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### Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...
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### Obstructions to Smith normal form of a special type

Suppose I have a Smith normal form $S,$ and I want to have an $M \in SL(n, \mathbb{Z}),$ such that $M - I$ has SNF $S.$ Is this always possible? For a (potentially) somewhat harder question, what if I ...
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### How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? ...
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### A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity: Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
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### Minkowski successive minima inequality for a lattice base?

Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors. The Minkowski successive minima inequality says ...
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### Linear independence over Q of logarithmic powers of prime numbers

I denote $p_k$ the $k^{th}$ prime number ($p_1=2$, etc...) Clearly, for any $n\in \mathbb{N}^*$, $(\log p_k)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$. My question concerns a ...
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### How many 0, 1 solutions would this system of underdetermined linear equations have?

The problem: I have a system of N linear equations, with K unknowns; and K > N. Although the equations are over $\mathbb Z$, the unknowns can only take the values 0 or 1. Here's an example with N=11 ...
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Let $p$ be a prime. Consider the following congruences: $$\begin{array}{lcl} a_1 x & = & c_1 (\text{mod } p) \\\\ \vdots & & \vdots\\\\ a_n x & = & c_n (\text{mod } p) \\ ... 0answers 194 views ### Algorithm/denominators of elements of a rational affine space I hope it's not a trivial question... Suppose I have a finite dimensional vector space V over \mathbb{Q} with a distinguished basis (in my case it's the kth graded piece of the free associative ... 3answers 986 views ### symmetric integer matrices Suppose I have a symmetric positive definite matrix M with integer entries. I want to decide whether M = A A^t, with A likewise integral. I assume that decision problem is NP-complete, as is the ... 0answers 402 views ### integral matrix of order p Hi everyone Let p be a prime number. I am interested to classify \{ A\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \} up to conjugacy. One reason to consider this problem is its relation to ... 6answers 2k views ### Status of the Hadamard Circulant conjecture The following feels like a community wiki question, so I do it here: Recently we have heard of a new proof of the Circulant Hadamard conjecture of Ryser (a long standing difficult conjecture): ... 0answers 641 views ### Inverse of a matrix with binomial coefficients Let a(n,k)=(-1)^k {{2n-k}\choose k} for 0 \le k \le n and a(n,k)=0 else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of (a(i,j))_{i,j\ge0} is ... 0answers 155 views ### Sum of two free o-submodules in a vector space over a local field Let V be a countably infinite dimensional K-vector space over a local field K (nontrivially discretely valued with finite residue field). Let o be the ring of integers of K. Given two free ... 0answers 329 views ### Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B? Let V be a finite dimensional linear space over a finite field k. Let A and B be two endomorphisms of V. Question 1. Is there an endomorphism C of V, which is expressed in terms of ... 0answers 565 views ### Regular languages of matrices and their generating functions My question is somewhat related to this question. Let us fix natural numbers k and C. Let A be an automaton whose alphabet consists of k\times k matrices with integer coefficients of ... 2answers 396 views ### A mapping from a lattice to itself Consider \mathbb{Z}^{n} for n = 2^r where r \geq 1 . Look at the iterates of the following function T from \mathbb{Z}^n to itself. T((a_1, a_2, \ldots, a_n)) = (|a_1 - a_n|, |a_2 - a_1|, ... 1answer 445 views ### Rational solutions of homogeneous equations Can every solution of a homogeneous linear system be approximated by a solution in rational numbers? In mathematical terms: Let$$Ax=0 be a homogeneous linear system in $n$ determinates for an ...
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This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
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### Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
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### Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below. Consider a second degree polynomial of two variables over the complex numbers. "P(x,y) = Ax^2 + Bxy + ...
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### Rational congruence of binomial coefficient matrices

Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on. Say that two matrices $A$ and $B$ over the ...
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### Prime/undecomposable matrices

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
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### Positive solutions of linear Diophantine equations

Let $A$ be a non-negative integer $k\times n$-matrix (i.e. each entry is non-negative and integer) with $rank(A) = k < n$. Let $b$ be a $k$-dimensional vector with positive integer entries. ...