Related papers: The density of integral quadratic forms having a $…

We show that the density of quadratic forms in $n$ variables over ${\mathbb Z}_p$ that are isotropic is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a…

We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When…

Let $K$ be a locally compact non-discrete field of characteristic $p>2$ and $Q$ be a non-degenerate isotropic binary quadratic form with coefficients in $K$. We obtain asymptotic estimates for the number of solutions in the two-fold product…

We study anisotropic universal quadratic forms over semi-global fields; i.e., over one-variable function fields over complete discretely valued fields. In particular, given a semi-global field $F$, we compute both the $m$-invariant of $F$…

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic +…

We complete a classification of quadratic forms over a field of characteristic 2 of type (1,3) that become isotropic over the function field of a quadric.

From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for…

We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…

We consider local densities for $p$-adic quaternion hermitian forms (hermitian forms over a division quaternion algebra over a ${\mathfrak p}$-adic field $k$). The author has studied such forms in connection with spherical functions on the…

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least…

An $n$-ary integral quadratic form is a formal expression $Q(x_1,...,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a poly$(n,k, \log p, \log t)$ randomized…

For an isotropic convex body $K\subset\mathbb{R}^n$ we consider the isotropic constant $L_{K_N}$ of the symmetric random polytope $K_N$ generated by $N$ independent random points which are distributed according to the cone probability…

Let $K$ be a convex body in $\R^d$, let $j\in\{1, ..., d-1\}$, and let $\varrho$ be a positive and continuous probability density function with respect to the $(d-1)$-dimensional Hausdorff measure on the boundary $\partial K$ of $K$. Denote…

The main result is the construction of ergodic transversal measures of full support on the space of all k-surfaces of a compact hyperbolic 3-manifold. This space is a laminated space, each of its leaf being identified with a "complete"…

Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of characteristic $\neq 2$, let $s$ be the unique non-negative integer such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$, and let $k$ denote the dimension of the anisotropic part…

We determine the probability that a random k-dimensional subspace of Euclidean n-space contains a positive vector.

For quadratic forms in $4$ variables defined over the rational function field in one variable over $\mathbb C(\!(t)\!)$, the validity of the local-global principle for isotropy with respect to different sets of discrete valuations is…

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy,…